Infinite Determinantal Measures and the Ergodic Decomposition of Infinite Pickrell Measures

Inﬁnite Determinantal Measures and The Ergodic Decomposition of Inﬁnite Pickrell Measures

A. Bufetov

REPORT No. 43, 2011/2012, spring ISSN 1103-467X ISRN IML-R- -43-11/12- -SE+spring INFINITE DETERMINANTAL MEASURES AND THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES

ALEXANDER I. BUFETOV

ABSTRACT. The main result of this paper, Theorem 1.11, gives an explicit description of the ergodic decomposition for infinite Pickrell measures on spaces of infinite complex matrices. The main construction is that of sigma-finite analogues of determinantal measures on spaces of configurations. An example is the infinite Bessel point process, the scaling limit of sigma-finite analogues of Jacobi orthogonal polynomial ensembles. The statement of Theorem 1.11 is that the infinite Bessel point process (subject to an appropriate change of variables) is precisely the ergodic decomposition measure for infinite Pickrell measures.

1. INTRODUCTION 1.1. Informal outline of the main results. The Pickrell family of measures is given by the formula (s) 2n s (1) µn = constn,s det(1 + z∗z)− − dz. Here n is a natural number, s a real number, z a square n n matrix with complex entries, dz the Lebesgue measure on the space of× such matrices, and constn,s a normalization constant whose precise choice will be (s) explained later. The measure µn is finite if s > 1 and infinite if s 1. (s) − ≤ − By definition, the measure µn is invariant under the actions of the unitary group U(n) by multiplication on the left and on the right. If the constants constn,s are chosen appropriately, then the Pickrell family of measures has the Kolmogorov property of consistency under natural (s) projections: the push-forward of the Pickrell measure µn+1 under the natural projection of cuttting the n n-corner of a (n + 1) (n + 1)-matrix × (s) × is precisely the Pickrell measure µn . This consistency property is also verified for infinite Pickrell measures provided n is sufficiently large; see Proposition 1.8 for the precise formulation. The consistency property and the Kolmogorov Theorem allow one to define the Pickrell family of measures µ(s), s R, on the space of infinite complex matrices. The Pickrell measures are∈ invariant by the action of the infinite unitary group on the left

Part of this work was done during a visit to the Institut Mittag-Leffler of the Royal Swedish Academy of Sciences 1 2 ALEXANDER I. BUFETOV and on the right, and the Pickrell family of measures is the natural analogue, in infinite dimension, of the canonical unitarily-invariant measure on a Grassmann manifold, see Pickrell [29]. What is the ergodic decomposition of Pickrell measures with respect to the action of the Cartesian square of the infinite unitary group? The ergodic unitarily-invariant probability measures on the space of infinite complex matrices admit an explicit classification due to Pickrell [27] and to which Olshanski and Vershik [26] gave a different approach: each ergodic measure is determined by an infinite array x = (x1, . . . , xn,... ) on the half- line, satisfying x1 x2 0 and x1 + + xn + < + , and an additional parameter≥γ˜ that· · · we ≥ call the Gaussian··· parameter··· . Informally,∞ the parameters xn should be thought of as “asymptotic singular values” of an infinite complex matrix, while γ˜ is the difference between the “asymptotic trace” and the sum of asymptotic eigenvalues (this difference is positive, in particular, for a Gaussian measure). Borodin and Olshanski [5] proved in 2000 that for finite Pickrell measures the Gaussian parameter vanishes almost surely, and the ergodic decomposition measure, considered as a measure on the space of configurations on the half-line (0, + ), coincides with the Bessel point process of Tracy and Widom [38], whose∞ correlation functions are given as determinants of the Bessel kernel. Borodin and Olshanski [5] posed the problem: Describe the ergodic decomposition of infinite Pickrell measures. This paper gives a solution to the problem of Borodin and Olshanski. The first step is the result of [9] that almost all ergodic components of an infinite Pickrell measure are themselves finite: only the decomposition measure itself is infinite. Furthermore, it will develop that, just as for finite measures, the Gaussian parameter vanishes. The ergodic decomposition measure can thus be identified with a sigma-finite measure B(s) on the space of configurations on the half-line (0, + ). How to describe a sigma-finite measure∞ on the space of configurations? Note that the formalism of correlation functions is completely unapplicable, since these can only be defined for a finite measure. This paper gives, for the first time, an explicit method for constructing infinite measures on spaces of configurations; since these measures are very closely related to determinantal probability measures, they are called infinite determinantal measures. We give three descriptions of the measure B(s); the first two can be carried out in much greater generality. 1. Inductive limit of determinantal measures. By definition, the measure B(s) is supported on the set of configurations X whose particles only accumulate to zero, not to infinity. B(s)-almost every configuration X thus THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 3 admits a maximal particle xmax(X). Now, if one takes an arbitrary R > 0 (s) and restricts the measure B onto the set X : xmax(X) < R , then the resulting restricted measure is finite and, after{ normalization, determinan-} tal. The corresponding operator is an orthogonal projection operator whose range is found explicitly for any R > 0. The measure B(s) is thus obtained as an inductive limit of finite determinantal measures along an exhausting family of subsets of the space of configurations. 2. A determinantal measure times a multiplicative functional. More generally, one reduces the measure B(s) to a finite determinantal measure by taking the product with a suitable multiplicative functional. A multiplicative functional on the space of configurations is obtained by taking the product of the values of a fixed nonnegative function over all particles of a configuration: Ψg(X) = g(x). x X Y∈ If g is suitably chosen, then the measure (s) (2) ΨgB is finite and, after normalization, determinantal. The corresponding operator is an orthogonal projection operator whose range is found explicitly. Of course, the previous description is a particular case of this one with g = χ(0,R). It is often convenient to take a positive function, for example, the function gβ(x) = exp( βx) for β > 0. While the range of the orthogonal projection operator inducing− the measure (2) is found explicitly for a wide class of functions g, it seems possible to give a formula for its kernel for only very few functions; these computations will appear in the sequel to this paper. 3. A skew-product with determinantal fibres. As was noted above, B(s)- almost every configuration X admits a maximal particle xmax(X), and it is natural to consider conditional measures of the measure B(s) with respect to the position of the maximal particle xmax(X). One obtains a well-defined determinantal probability measure induced by a projection operator whose range, again, is found explicitly using the description of Palm measures of determinantal point processes due to Shirai and Takahashi [34] . The sigma-finite “distribution” of the maximal particle is also explicitly found: the ratios of the measures of intervals are obtained as ratios of suitable Fred- holm determinants. The measure B(s) is thus represented as a skew-product whose base is an explicitly found sigma-finite measure on the half-line, and whose fibres are explicitly found determinantal probability measures. See section 1.10 for a detailed presentation. The key roleˆ in the construction of infinite determinantal measures is played by the result of [10] (see also [11]) that a determinantal probability 4 ALEXANDER I. BUFETOV measure times an integrable multiplicative functional is, after normalization, again a determinantal probability measure whose operator is found explicitly. In particular, if PΠ is a determinantal point process induced by a projection operator Π with range L, then, under certain additional assumptions, the measure ΨgPΠ is, after normalization, a determinantal point process induced by the projection operator onto the subspace √gL. Informally, if g is such that the subspace √gL no longer lies in L2, then the measure ΨgPΠ ceases to be finite, and one obtains, precisely, an infinite determinantal measure corresponding to a subspace of locally square- integrable functions, one of the main constructions of this paper, see Theo- rem 2.11. The Bessel point process of Tracy and Widom, which governs the ergodic decomposition of finite Pickrell measures, is the scaling limit of Jacobi orthogonal polynomial ensembles. In the problem of ergodic decomposition of infinite Pickrell measures one is led to investigating the scaling limit of infinite analogues of Jacobi orthogonal polynomial ensembles. The resulting scaling limit, an infinite determinantal measure, is computed in the paper and called the infinite Bessel point process; see Subsection 1.4 of this Introduction for the precise definition. The main result of the paper, Theorem 1.11, identifies the ergodic decomposition measure of an infinite Pickrell measure with the infinite Bessel point process.

1.1.1. The Jacobi Orthogonal Polynomial Ensemble. Take n N, s R, and endow the cube ( 1, 1)n with the measure ∈ ∈ −

n (3) (u u )2 (1 u )sdu . i − j − i i 1 i 1, the measure (3) is a particular case of the Jacobi orthogonal polynomial− ensemble. Under the scaling y (4) u = 1 i , i = 1, . . . , n, i − 2n2 the ensemble (3) converges to the Bessel point process of Tracy and Widom [38], a determinantal point process with the kernel

√y1Js+1(√y1)Js(√y2) √y2Js+1(√y2)Js(√y1) (5) J˜ (y , y ) = − s 1 2 2(y y ) 1 − 2 (see, e.g., page 295 in Tracy and Widom [38]); recall that the Bessel point process is governed by the projection operator, in L2((0, + ), Leb), onto the subspace of functions whose Hankel transform is supported∞ in [0, 1]. THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 5

For s 1, the measure (3) is infinite. Our first aim is to introduce the infinite Bessel≤ − point process, the scaling limit of the measures (3) under the scaling (4).

1.1.2. The recurrence relation for Jacobi orthogonal polynomial ensembles. In order to motivate the construction of inﬁnite determinantal measures, we start with the recurrence relation for Jacobi orthogonal polyno- (s) mial ensembles Let s > 1, let Pn be the standard Jacobi orthogonal − s (s) polynomials corresponding to the weight (1 u) . Let K˜n (u1, u2) the n-th Christoffel-Darboux kernel of the Jacobi orthogonal− polynomial ensemble. We can now represent the measure (3) in determinantal form: (6) n 1 n const (u u )2 (1 u )sdu = det K˜ (s)(u , u ) du , n,s i − j − i i n! n i j · i 1 i

(s,n) s/2 s/2 s/2 n 1 (7) L = span((1 u) , (1 u) u, . . . , (1 u) u − ) = Jac − − − s/2 s/2+1 s/2+n 1 = span((1 u) , (1 u) ,..., (1 u) − ). − − − By deﬁnition, we have a direct-sum decomposition

(s,n) s/2 (s+2,n 1) L = C(1 u) L − Jac − ⊕ Jac A straightforward veriﬁcation yields, for any s > 1, the recurrence relation −

˜ (s) s + 1 (s+1) s/2 (s+1) s/2 (8) Kn (u1, u2) = Pn 1 (u1)(1 u1) Pn 1 (u2)(1 u2) + 2s+1 − − − − ˜ (s+2) + Kn (u1, u2) and, consequently, an orthogonal direct-sum decomposition

(s,n) (s+1) s/2 (s+2,n 1) LJac = CPn 1 (u)(1 u) LJac − . − − ⊕ 6 ALEXANDER I. BUFETOV

1.1.3. The analogue of the recurrence relation for s 1. For s 1, s ≤1 1 ≤ − deﬁne a natural number ns by the relation 2 + ns ( 2 , 2 ] and introduce the subspace ∈ − (9) (s,n) s/2 s/2+1 (s+2ns 1) s/2+n 1 ˜ − s V = span((1 u) , (1 u) ,...,Pn ns (u)(1 u) − ). − − − − By deﬁnition, we have is a direct sum decomposition

(s,n) (s,n) (s+2ns,n ns) (10) L = V˜ L − . Jac ⊕ Jac Note that

(s+2ns,n ns) (s,n) L − L ([ 1, 1], Leb), V˜ L ([ 1, 1], Leb) = 0. Jac ⊂ 2 − 2 − \ 1.1.4. Scaling limits. Recall that the scaling limit, with respect to the scal- (s) ing 4), of Christoffel-Darboux kernels K˜n of the Jacobi orthogonal polynomial ensemble, is given by the Bessel kernel J˜s of Tracy and Widom [38]. It is clear that, for any β, under the scaling (4), we have

2 β β β lim (2n ) (1 ui) = yi n →∞ − and, for any α > 1, by the classical Heine-Mehler asymptotics for Jacobi polynomials, we have−

α+1 α 1 Jα(√yi) 2 1 (α) −2 lim 2− n− Pn (ui)(1 ui) = . n →∞ − √yi It is therefore natural to take the subspace

Js+2ns 1(√y) (11) V˜ (s) = span(ys/2, ys/2+1,..., − ). √y as the scaling limit of the subspace (9). Furthermore, we already know that the scaling limit of the subspace (10) ˜(s+2ns) ˜ is the subspace L , the range of the operator Js+2ns . This motivates introducing the subspace H˜ (s)

(12) H˜ (s) = V˜ (s) L˜(s+2ns). ⊕ It is natural to consider the subspace H˜ (s) as the scaling limit of the subspaces L(s,n) under the scaling (4) as n . Jac → ∞ Note that the subspace H˜ (s) consists of locally square-integrable functions, which, moreover, only fail to be square-integrable at zero: for any (s) ε > 0, the subspace χ[ε,+ )H˜ is contained in L2. ∞ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 7

1.1.5. Definition of the infinite Bessel point process. We now proceed to a precise description, in this specific case, of one of the main constructions of the paper: that of a sigma-finite measure B˜ (s), the scaling limit of infinite Jacobi ensembles (3) under the scaling (4). Let Conf((0, + )) be the space ∞ of configurations on (0, + ). Given a Borel subset E0 (0, + ), we let Conf((0, + ),E ) be the∞ subspace of configurations all⊂ whose∞ particles ∞ 0 lie in E0. Generally, given a measure B on a set X and a measurable subset Y X such that 0 < B(Y ) < + , we let B Y stand for the restriction of ⊂ ∞ | the measure B onto the subset Y . It will be proved in what follows that, for any ε > 0, the subspace (s) χ(ε,+ )H˜ is a closed subspace of L2((0, + ), Leb) and that the operator ∞ (ε,s ∞ (s) Π˜ ) of orthogonal projection onto the subspace χ(ε,+ )H˜ is locally of ∞ trace class. By the Macch`ı-Soshnikov Theorem, the operator Π˜ (ε,s) induces a determinantal measure P ˜ (ε,s on Conf((0, + )). Π ) ∞ Proposition 1.1. Let s 1. There exists a sigma-finite measure B˜ (s) on Conf((0, + )) such that≤ we − have ∞ (1) the particles of B˜ (s)-almost every configuration do not accumulate at zero; (2) for any ε > 0 we have

0 < B˜ (s)(Conf((0, + ); (ε, + )) < + ∞ ∞ ∞ and ˜ (s) B Conf((0,+ );(ε,+ )) ∞ ∞ | = PΠ˜ (ε,s) . B˜ (s)(Conf((0, + ); (ε, + )) ∞ ∞ These conditions define the measure B˜ (s) uniquely up to multiplication by a constant. Remark. For s = 1, 3,... , we can also write 6 − − (s) s/2 s/2+ns 1 (s+2ns) H˜ = span(y , . . . , y − ) L˜ ⊕ and use the preceding construction otherwise without change. For s = 1 note that the function y1/2 fails to be square-integrable at infinity — whence− ˜ (s) the need for the definition given above. For s > 1, write B = P ˜ . − Js ˜ (s1) ˜ (s2) Proposition 1.2. If s1 = s2, then the measures B and B are mutully singular. 6 Proposition 1.2 will be derived from Proposition 1.4 below, which in turn, will be obtained as a corollary of the main result, Theorem 1.11, in the last section of the paper. 8 ALEXANDER I. BUFETOV

1.2. The modiﬁed Bessel point process. In what follows, we will need the Bessel point process subject to the change of variable y = 4/x. We thus consider the half-line (0, + ) endowed with the standard Lebesgue ∞ measure Leb. Take s > 1 and introduce a kernel J (s) by the formula − 2 1 2 2 1 2 Js( √x ) √x Js+1( √x ) Js( √x ) √x Js+1( √x ) (13) J (s)(x , x ) = 1 2 2 − 2 1 1 , 1 2 x x 1 − 2 x1 > 0, x2 > 0 . or, equivalently,

1 (s) 1 t t (14) J (x, y) = Js(2 )Js(2 )dt. x1x2 x1 x2 Z0 r r The change of variable y = 4/x reduces the kernel J (s) to the kernel J˜s of the Bessel point process of Tracy and Widom considered above (recall here that a change of variables u1 = ρ(v1), u2 = ρ(v2) transforms a kernel K(u1, u2) to a kernel of the form K(ρ(v1), ρ(v2))( ρ0(v1)ρ0(v2))). The kernel J (s) therefore induces on the space L ((0, + ), Leb) a locally 2 p trace class operator of orthogonal projection, for which,∞ slightly abusing notation, we keep the symbol J (s); we denote L(s) the range of J (s). By the Macch`ı-Soshnikov Theorem, the operator J (s) induces a determinantal measure PJ(s) on Conf((0, + )). ∞ 1.3. The modified infinite Bessel point process. The involutive homeomorphism y = 4/x of the half-line (0, + ) induces a corresponding ∞ change of variable homeomorphism of the space Conf((0, + )). Let B(s) ∞ be the image of B˜ (s) under our change of variables. As we shall see below, the measure B(s) is precisely the ergodic decomposition measure for the infinite Pickrell measures. A more explicit description description of the measure B(s) can be given as follows. By definition, we have L(s) = ϕ(4/x) , ϕ L˜(s) . We similarly let V (s),H(s) L ((0, + ), Leb) be { x ∈ } ⊂ 2,loc ∞ the images of V˜ (s), H˜ (s) under our change of variables y = 4/x: ϕ(4/x) ϕ(4/x) V (s) = , ϕ V˜ (s) ,H(s) = , ϕ H˜ (s) . { x ∈ } { x ∈ }

2 Js+2ns 1( ) (s) s/2 1 − √x (s) By definition, we have V = span(x− − ,..., √x ); H = V (s) L(s+2ns). ⊕ (s) It will develop that for all R > 0 the subspace χ(0,R)H is a closed subspace in L ((0, + ), Leb); let Π(s,R) be the corresponding orthogonal 2 ∞ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 9 projection operator. By definition, the operator Π(s,R) is locally of trace- class and, by the Macch`ı-Soshnikov Theorem, the operator Π(s,R) induces a determinantal measure PΠ(s,R) on Conf((0, + )). ∞ The measure B(s) is characterized by the following conditions: (1) the set of particles of B(s)-almost every configuration is bounded; (2) for all R > 0 we have 0 < B(Conf((0, + ); (0,R)) < + and ∞ ∞ B Conf((0,+ );(0,R)) | ∞ = PΠ(s,R) . B(Conf((0, + ); (0,R)) ∞ These conditions define the measure B(s) uniquely up to multiplication by a constant. Remark. For s = 1, 3,... , we can of course also write 6 − − (s) s/2 1 s/2 ns+1 (s+2ns) H = span(x− − , . . . , x− − ) L . ⊕ Let I1,loc((0, + ), Leb) be the space of locally trace-class operators acting on the space L ((0∞ , + ), Leb) We have the following proposition describ- 2 ∞ ing the asymptotic behaviour of the operators Π(s,R) as R . → ∞ Proposition 1.3. Let s 1. Then ≤ − (1) as R we have Π(s,R) J (s+2ns) in I ((0, + ), Leb); → ∞ → 1,loc ∞ (2) Consequently, as R , we have PΠ(s,R) PJ(s+2ns) weakly in the space of probability→ measures∞ on Conf((0→, + )). ∞ (s) As before, for s > 1, write B = PJ(s) . Proposition 1.2 is equivalent to the following −

(s1) (s2) Proposition 1.4. If s1 = s2, then the measures B and B are mutully singular. 6 Proposition 1.4 will be obtained as the corollary of the main result, The- orem 1.11, in the last section of the paper. We now represent the measure B(s) as the product of a determinantal probability measure and a multiplicative functional. Here we limit ourselves to speciﬁc example of such a representation, but in what follows we will see that they can be constructed in much greater generality. Introduce a function S on the space Conf((0, + )) by setting ∞ S(X) = x. x X X∈ The function S may, of course, assume value , but the set of such conﬁg- ∞ urations is B(s)-negligible, as is shown by the following Proposition 1.5. For any s R we have S(X) < + almost surely with ∈ ∞ respect to the measure B(s) and for any β > 0 we have (s) exp( βS(X)) L1(Conf((0, + )), B ). − ∈ ∞ 10 ALEXANDER I. BUFETOV

Furthermore, we shall now see that the measure

exp( βS(X))B(s) − exp( βS(X))dB(s) Conf((0,+ )) − R ∞ is determinantal.

Proposition 1.6. For any s R, β > 0, the subspace ∈ (15) exp( βx/2)H(s) − is a closed subspace of L2 (0, + ), Leb , and the operator of orthogonal projection onto the subspace (15)∞ is locally of trace class. Let Π(s,β) be the operator of orthogonal projection onto the subspace (15). By Proposition 1.6 and the Macch`ı-Soshnikov Theorem, the operator Π(s,β) induces a determinantal probability measure on the space Conf((0, + )). ∞ Proposition 1.7. For any s R, β > 0, we have ∈ exp( βS(X))B(s) (16) − = PΠ(s,β) . exp( βS(X))dB(s) Conf((0,+ )) − R ∞ 1.4. Unitarily-Invariant Measures on Spaces of Inﬁnite Matrices.

1.4.1. Pickrell Measures. Let Mat(n, C) be the space of n n matrices with complex entries: ×

Mat(n, C) = z = (zij), i = 1, . . . , n; j = 1, . . . , n { } Let Leb = dz be the Lebesgue measure on Mat(n, C). For n1 < n, let n πn : Mat(n, C) Mat(n1, C) 1 → be the natural projection map that to a matrix z = (zij), i, j = 1, . . . , n, n assigns its upper left corner, the matrix πn1 (z) = (zij), i, j = 1, . . . , n1. (s) Following Pickrell [27], take s R and introduce a measure µn on ∈ Mat(n, C) by the formula

(s) 2n s e µn = det(1 + z∗z)− − dz.

(s) The measure µn is ﬁnite if and only if s > 1. (s) e − The measures µn have the following property of consistency with re- n spect to the projectionse πn1 . e THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 11

Proposition 1.8. Let s R, n N satisfy n + s > 0. Then for any ∈ ∈ z˜ Mat(n, C) we have ∈

2n 2 s (17) det(1 + z∗z)− − − dz =

n+1Z 1 (πn )− (˜z) 2n+1 2 π (Γ(n + 1 + s)) 2n s = det(1 +z ˜∗z˜)− − . Γ(2n + 2 + s) Γ(2n + 1 + s) · Now let Mat(N, C) be the space of inﬁnite matrices whose rows and columns are indexed by natural numbers and whose entries are complex:

Mat(N, C) = z = (zij), i, j N, zij C . { ∈ ∈ } Let πn∞ : Mat(N, C) Mat(n, C) be the natural projection map that to → an infinite matrix z Mat(N, C) assigns its upper left n n-“corner”, the ∈ × matrix (zij), i, j = 1, . . . , n. For s > 1, Proposition 1.8 together with the Kolmogorov Existence Theorem [19]− implies that there exists a unique probability measure µ(s) on Mat(N, C) such that for any n N we have the relation ∈ n (s) n2 Γ(2l + s)Γ(2l 1 + s) (s) (πn∞) µ = π− − µn . ∗ (Γ(l + s))2 Yl=1 If s 1, then Proposition 1.8 together with the Kolmogorove Existence Theorem≤ − [19] implies that for any λ > 0 there exists a unique infinite measure µ(s,λ) on Mat(N, C) such that (1) for any n N satisfying n + s > 0 and any compact subset Y ∈ (s,λ) (s,λ⊂) Mat(n, C) we have µ (Y ) < + ; the pushforwards (πn∞) µ are consequently well-defined; ∞ ∗ (2) for any n N satisfying n + s > 0 we have ∈ n (s,λ) 2n Γ(2l + s)Γ(2l 1 + s) (s) (18) (πn∞) µ = λ( π− − )µ . ∗ (Γ(l + s))2 lY=n0 The measures µ(s,λ) will be called infinite Pickrell measurese . Slightly abusing notation, we shall omit the super-script λ and write µ(s) for a measure defined up to a multiplicative constant. See p.116 in Borodin and Ol- shanski [5] for a detailed presentation of infinite Pickrell measures.

(s1) Proposition 1.9. For any s1, s2 R, s1 = s2, the Pickrell measures µ ∈ 6 and µ(s2) are mutually singular. Proposition 1.9 is obtained from Kakutani’s Theorem in the spirit of [5], see also [24]. 12 ALEXANDER I. BUFETOV

Let U( ) be the infinite unitary group: an infinite matrix u = (uij)i,j N belongs to∞U( ) if there exists a natural number n such that the matrix∈ ∞ 0 (u ), i, j [1, n ] ij ∈ 0 is unitary, while u = 1 if i > n and u = 0 if i = j, max(i, j) > n . ii 0 ij 6 0 The group U( ) U( ) acts on Mat(N, C) by multiplication on both sides: ∞ × ∞ 1 T(u1,u2)z = u1zu2− . The Pickrell measures µ(s) are by definition U( ) U( )-invariant. For the roleˆ of Pickrell and related measures in the representation∞ × ∞ theory of U( ), see [24] [26], [7]. The∞ main result of this paper is an explicit description of the measure µ(s) and its identification, after a change of variable, with the infinite Bessel point process considered above.

1.5. Classiﬁcation of ergodic measures. First, we recall the classiﬁcation of ergodic probability U( ) U( )-invariant measures on Mat(N, C) (see, e.g., Pickrell [27], [28],∞ × Olshanski∞ and Vershik [26], Rabaoui [30], [31]). (n) Take z Mat(N, C), denote z = πn∞z, and let ∈ (n) (n) (19) λ1 > ... > λn > 0 (n) (n) be the eigenvalues of the matrix (z )∗z , counted with multiplicities, arranged in non-increasing order. To stress dependence on z, we write λ(n) = λ(n)(z). If η be an ergodic Borel U( ) U( )-invariant prob- i i ∞ × ∞ ability measure on Mat(N, C), then there exist non-negative real numbers

γ > 0, x1 > x2 > ... > xn > ... > 0 ,

∞ satisfying γ > xi, such that for η-almost every z Mat(N, C) and any i=1 ∈ i N we have: ∈ P (n) (n) (n) λi (z) tr(z )∗z (20) xi = lim , γ = lim . n 2 n 2 →∞ n →∞ n Conversely, given non-negative real numbers γ > 0, x1 > x2 > ... > xn > ... > 0 such that ∞ γ > xi , i=1 X there exists a unique U( ) U( )-invariant ergodic Borel probability ∞ × ∞ measure η on Mat(N, C) such that the relations (20) hold for η-almost all z Mat(N, C). ∈ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 13

N Introduce the Pickrell set ΩP R+ R+ by the formula ⊂ × ∞ ΩP = ω = (γ, x): x = (xn), n N, xn xn+1 0, γ xi . { ∈ > > > } i=1 X N The set ΩP is, by deﬁnition, a closed subset of R+ R+ endowed with the Tychonoff topology. For ω Ω we let η be the× corresponding ergodic ∈ P ω probability measure. The Fourier transform of the measure ηω is explicitly described as follows. First, for any λ R we have ∈ ∞ 2 exp( 4(γ xk)λ ) − − k=1 (21) exp(iλ z11)dηω(z) = . < ∞ P 2 Mat(ZN,C) (1 + 4xkλ ) k=1 Q Denote Fω(λ) the expression in the right-hand side of (21); then, for any λ1, . . . , λm R we have (22) ∈ exp(i(λ z + + λ z ))dη (z) = F (λ ) F (λ ). 1< 11 ··· m< mm ω ω 1 ····· ω m Mat(ZN,C)

The Fourier transform is fully defined, and the measure ηω is completely described. An explicit construction of the ergodic measures ηω is given as follows. First, if one takes all entries of the matrix z are independent identically distributed complex Gaussian random variables with expectation 0 and variance γ˜, then the resulting Gaussian measure with parameter γ˜, clearly unitarily invariant and, by the Kolmogorov zero-one law, ergodic, corresponds to the parameter ω = (˜γ, 0,..., 0,... ) — all x-coordinates are equal to 0 ( indeed, singular values of a Gaussian matrix grow at rate √n rather than n). Next, let (v1, . . . , vn,... ), (w1, . . . , wn,... ) be two infinite independent vectors of independent identically distributed complex Gaussian random variables with variance √x, and set zij = viwj. One thus obtains a measure whose unitary invariance is clear and whose ergodicity is immediate from the Kolmogorov zero-one law. This measure corresponds to the parameter ω ΩP such that γ(ω) = x, x1(ω) = x, and all the other parameters are∈ zero. Following Olshanski and Vershik [26], such measures are called ∞ Wishart measures with parameter x. In the general case, set γ˜ = γ xk. − k=1 The measure ηω is then an infinite convolution of the Wishart measuresP with parameters x1, . . . , xn,... and the Gaussian measure with parameter γ˜. Convergence of the series x1 + +xn +... ensures that the convolution is well-defined. ··· 14 ALEXANDER I. BUFETOV

∞ The quantity γ˜ = γ xk will therefore be called the Gaussian param- −k=1 eter of the measure ηω. ItP will develop that the Gaussian parameter vanishes for almost all ergodic components of Pickrell measures. By Proposition 3 in [8], the subset of ergodic U( ) U( )-invariant measures is a Borel subset of the space of all Borel probability∞ × ∞ measures on Mat(N, C) endowed with the natural Borel structure (see, e.g., [2]). Fur- thermore, if one denotes ηω the Borel ergodic probability measure corresponding to a point ω Ω , ω = (γ, x), then the correspondence ∈ P ω η −→ ω is a Borel isomorphism of the Pickrell set Ω and the set of U( ) U( )- P ∞ × ∞ invariant ergodic probability measures on Mat(N, C). The Ergodic Decomposition Theorem (Theorem 1 and Corollary 1 of [8]) implies that each Pickrell measure µ(s), s R, induces a unique decompos- (s) ∈ ing measure µ on ΩP such that we have

(s) (s) (23) µ = ηω dµ (ω) .

ΩZP The integral is understood in the usual weak sense, see [8]. For s > 1, the measure µ(s) is a probability measure on Ω , while for − P s 6 1 the measure µ(s) is infinite. Set− ∞ Ω0 = (γ, x ) Ω : x > 0 for all n, γ = x . P { { n} ∈ P n n} n=1 X 0 The subset ΩP is of course not closed in ΩP . Introduce a map conf : Ω Conf((0, + )) P → ∞ that to a point ω Ω , ω = (γ, x ) assigns the configuration ∈ P { n} conf(ω) = (x , . . . , x ,...) Conf((0, + )). 1 n ∈ ∞ The map ω conf(ω) is bijective in restriction to the subset Ω0 . → P 1.6. Formulation of the main result. We start by formulating the analogue of the Borodin-Olshanski Ergodic Decomposition Theorem [5] for finite Pickrell measures. Proposition 1.10. Let s > 1. Then µ(s)(Ω0 ) = 1 and the µ(s)-almost sure − P bijection ω conf(ω) identifies the measure µ(s) with the determinantal → measure PJ(s) . The main result of this paper, an explicit description for the ergodic decomposition of infinite Pickrell measures, is given by the following THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 15

Theorem 1.11. Let s R, and let µ(s) be the decomposing measure, defined by (23), of the Pickrell∈ measure µ(s). Then (1) µ(s)(Ω Ω0 ) = 0; P \ P (2) the µ(s)-almost sure bijection ω conf(ω) identifies µ(s) with the → infinite determinantal measure B(s). 1.7. A skew-product representation of the measure B(s). With respect to the measure B(s), almost every configuration X only accumulates at zero and therefore admits a maximal particle that we denote xmax(X). We are interested in the distribution of the maximal particle under the measure B(s). By definition, for any R > 0, the measure B(s) assigns finite weight to the set X : xmax(X) < R . Furthermore, again by definition, for any R > 0 and{R ,R R we have} the following relation: 1 2 ≤ (s) (s,R) B ( X : xmax(X) < R1 ) det(1 χ(R1,+ )Π χ(R1,+ )) ∞ ∞ (24) (s) { } = − (s,R) . B ( X : xmax(X) < R2 ) det(1 χ(R2,+ )Π χ(R2,+ )) { } − ∞ ∞ The push-forward of the measure B(s) is a well-defined Borel sigma-finite (s) measure on (0, + ) for which we will use the symbol ξmaxB ; the mea- (s) ∞ sure ξmaxB is, of course, defined up to multiplication by a positive constant. (s) Question. What is the asymptotics of the quantity ξmaxB (0,R) as R ? as R 0? → ∞The operator→ Π(s,R) admits a kernel for which we keep the same sym- (s,R) bol; consider the function ϕR(x) = Π (x, R). By definition, ϕR(x) (s) (s,R) ∈ χ(0,R)H . Let H stand for the orthogonal complement to the one- (s) dimensional subspace spanned by ϕR(x) in χ(0,R)H . In other words, (s,R) (s) H is the subspace of those functions in χ(0,R)H that assume value (s,R) zero at the point R. Let Π be the operator of orthogonal projection onto (s,R) the subspace H .

(s) ∞ (s) = (s,R) dξ (R). Proposition 1.12. We have B PΠ maxB 0 R Proof. This immediately follows from the deﬁnition of the measure B(s) and the characterization of Palm measures for determinantal point processes due to Shirai and Takahashi [33].

1.8. The general scheme of ergodic decomposition.

1.8.1. Approximation. Let F be the family of σ-inﬁnite U( ) U( )- ∞ × ∞ invariant measures µ on Mat(N, C) for which there exists n0 (dependent on 16 ALEXANDER I. BUFETOV

µ) such that for all R > 0 we have

µ( z : max zij < R ) < + . { 16i,j6n0 | | } ∞ By definition, all Pickrell measures belong to the class F. We recall the result of [9] stating that every ergodic measure belonging to the class F must be finite and that the ergodic components of any measure in F are therefore almost surely finite (the existence of the ergodic decomposition for any measure µ F follows from the ergodic decomposition theorem for actions of inductively∈ compact groups established in [8]). The classification of finite ergodic measures now implies that for every measure µ F there ∈ exists a unique Borel σ-finite measure µ on the Pickrell set ΩP such that

(25) µ = ηω dµ(ω).

ΩZP Our next aim is to construct, following Borodin and Olshanski [5], a sequence of finite-dimensional approximations for the measure µ. To a matrix z Mat(N, C) and a number n N assign the array ∈ ∈ (n) (n) (n) (λ1 , λ2 , . . . , λn ) (n) (n) of eigenvalues arranged in non-increasing order of the matrix (z )∗z , where (n) z = (zij)i,j=1,...,n. For n N define a map ∈ (n) r : Mat(N, C) ΩP → by the formula (n) (n) (n) (n) 1 (n) (n) λ1 λ2 λn (26) r (z) = ( tr(z )∗z , , ,..., , 0, 0,...). n2 n2 n2 n2 It is clear by definition that for any n N, z Mat(N, C) we have ∈ ∈ r(n)(z) Ω0 . ∈ P For any µ F and all sufficiently large n N the push-forwards (r(n)) µ are well-defined∈ since the unitary group is compact.∈ We shall presently see∗ that for any µ F the measures (r(n)) µ approximate the ergodic decomposition measure∈ µ. ∗ We start by a direct description of the map that takes a measure µ F to its ergodic decomposition measure µ. ∈ Following Borodin-Olshanski [5], let Matreg(N, C) be the set of all matrices z such that 1 (k) (1) for any k, there exists the limit lim 2 λn =: xk(z); n n →∞ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 17

1 (n) (n) (2) there exists the limit lim 2 tr(z )∗z =: γ(z). n n →∞ Since the set of regular matrices has full measure with respect to any finite ergodic U( ) U( )-invariant measure, the existence of the ergodic ∞ × ∞ decomposition (25) implies µ(Mat(N, C) Matreg(N, C)) = 0. We intro- ( ) duce the map r ∞ : Matreg(N, C) ΩP by the formula → ( ) r ∞ (z) = (γ(z), x1(z), x2(z), . . . , xk(z),...). The Ergodic Decomposition Theorem [8] and the classification of ergodic unitarily-invariant measures in the form of Olshanski and Vershik [26] imply the important equality ( ) (27) (r ∞ ) µ = µ. ∗ Given a complete separable metric space Z, we write Mfin(Z) for the space of all finite Borel measures on Z endowed with the weak topology. Recall [2] that Mfin(Z) is itself a complete separable metric space: the weak topology is induced, for instance, by the Levy-Prohorov´ metric. We proceed to showing that the measures (r(n)) µ approximate the mea- ( ) ∗ sure (r ∞ ) µ = µ as n . For finite measures µ the following statement is due to Borodin∗ and Olshanski→ ∞ [5]. Proposition 1.13. Let µ be a finite σ-invariant measure on Mat(N, C). Then, as n , we have → ∞ (n) ( ) (r ) µ (r ∞ ) µ ∗ → ∗ weakly in Mfin(ΩP ).

Proof. Let f :ΩP R be continuous and bounded. For any z → (n) ( ) ∈ Matreg(N, C), by deﬁnition, we have r (z) r ∞ (z) as n , and, consequently, also, → → ∞ (n) ( ) lim f(r (z)) = f(r ∞ (z)), n →∞ whence, by bounded convergence theorem, we have

(n) ( ) lim f(r (z)) dµ(z) = f(r ∞ (z)) dµ(z). n →∞ Mat(ZN,C) Mat(ZN,C) Changing variables, we arrive at the convergence

(n) ( ) lim f(ω) d(r ) µ = f(ω) d(r ∞ ) µ, n ∗ ∗ →∞ ΩZP ΩZP and the desired weak convergence is established. For σ-ﬁnite measures µ F, the Borodin-Olshanski proposition is mod- iﬁed as follows. ∈ 18 ALEXANDER I. BUFETOV

Lemma 1.14. Let µ F. There exists a positive bounded continuous func- ∈ tion f on the Pickrell set ΩP such that ( ) (n) (1) f L1(ΩP , (r ∞ ) µ) and f L1(ΩP , (r ) µ) for all sufﬁciently ∈ ∗ ∈ ∗ large n N; (2) as n ∈ , we have → ∞ (n) ( ) f(r ) µ f(r ∞ ) µ ∗ → ∗ weakly in Mﬁn(ΩP ). Proof of Lemma 1.14 will be given in Section 7.

1.8.2. Convergence of probability measures on the Pickrell set. Recall that we have a natural forgetting map conf : Ω Conf(0, + ) that to a point P → ∞ ω = (γ, x), x = (x1, . . . , xn,... ), assigns the configuration conf(ω) = (x1, . . . , xn,... ). For ω Ω , ω = (γ, x), x = (x , . . . , x ,... ), x = x (ω), set ∈ P 1 n n n ∞ S(ω) = xn(ω). n=1 X In other words, we set S(ω) = S(conf(ω)), and, slightly abusing notation, keep the same symbol for the new map. Take β > 0 and consider the measures exp( βS(ω))r(n)(µ(s))), n N. − ∈ Proposition 1.15. For any s R, β > 0, we have ∈ exp( βS(ω)) L (Ω , r(n)(µ(s))). − ∈ 1 P Introduce the probability measure exp( βS(ω))r(n)(µ(s)) ν(s,n,β) = − . exp( βS(ω))dr(n)(µ(s)) ΩP − R Now go back to the determinantal measure PΠ(s,β) on the space Conf((0, + )) (s,β) ∞ (cf. (16)) and let the measure ν on ΩP be defined by the requirements (s,β) 0 (1) ν (ΩP ΩP ) = 0; (s,β\) (2) conf ν = PΠ(s,β) . ∗ The key roleˆ in the proof of Theorem 1.11 is played by Proposition 1.16. For any β > 0, s R, as n we have ∈ → ∞ ν(s,n,β) ν(s,β) → weakly in the space Mfin(ΩP ). THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 19

Proposition 1.16 will be proved in Section 6, and in Section 7, using Proposition 1.16, combined with Lemma 1.14, we will conclude the proof of the main result, Theorem 1.11. To establish weak convergence of the measures ν(s,n,β), we first study scaling limits of the radial parts of finite-dimensional projections of infinite Pickrell measures.

1.9. The radial part of the Pickrell measure. Following Pickrell, to a matrix z Mat(n, C) assign the collection (λ1(z), . . . , λn(z)) of the eigen- ∈ values of the matrix z∗z arranged in non-increasing order. Introduce a map

n radn : Mat(n, C) R+ → by the formula (28) rad : z (λ (z), . . . , λ (z)). n → 1 n

The map (28) naturally extends to the map radn πn∞ defined on Mat(N, C); ◦ we keep the same symbol radn for the extended map. The radial part of (s) the Pickrell measure µn is now defined as the push-forward of the mea- (s) sure µn under the map radn. Note that, since finite-dimensional unitary groups are compact, and, by definition, for any s and all sufficiently large (s) n, the measure µn assigns finite weight to compact sets, the pushforward is well-defined, for sufficiently large n, even if the measure µ(s) is infinite. Slightly abusing notation, we write dz for the Lebesgue measure on n Mat(n, C) and dλ for the Lebesgue measure on R+. For the push-forward of the Lebesgue measure Leb(n) = dz under the map radn we now have

2 (radn) (dz) = const(n) (λi λj) dλ, ∗ · − i

(s) 2 1 (29) (radn) µn = const(n, s) (λi λj) dλ, ∗ · − · (1 + λ )2n+s i

xi 1 (30) ui = − . xi + 1 20 ALEXANDER I. BUFETOV

(s) Proposition 1.17. In the coordinates (30) the radial part (radn) µn of the ∗ (s) n measure µn is defined on the cube [ 1, 1] by the formula − n (s) 2 s (31) (radn) µn = const(n, s) (ui uj) (1 ui) dui. ∗ · − · − i 1, the constant const(n, s) can be chosen in such a way that the right-hand− side be a probability measure; in the case s 1, there is no canonical normalization, the left hand side is defined up to≤ proportion- − ality, and a positive constant can be chosen arbitrarily. For s > 1, Proposition 1.17 yields a determinantal representation for the radial part− of the Pickrell measure: namely, the radial part is identified with the Jacobi orthogonal polynomial ensemble in the coordinates (30). Passing to the scaling limit, one obtains the Bessel point process (subject to the change of variable y = 4/x). Similarly, it will develop that for s 1, the scaling limit of the measures (31) is precisely the modified infinite≤ − Bessel point process introduced above. Furthermore, if one multiplies the measures (31) by the density exp( βS(X)/n2), then the resulting measures are finite and determinantal, and their− weak limit, after approproate scaling, is precisely the determinantal measure PΠ(s,β) of (16). This weak convergence is a key step in the proof of Proposition 1.16. The study of the case s 1 thus requires a new object: infinite determinantal measures on spaces≤ − of configurations. In the next Section, we proceed to the general construction and description of the properties of infinite determinantal measures. Acknowledgements. Grigori Olshanski posed the problem to me, and I am greatly indebted to him. I am deeply grateful to Alexei Borodin, Alexei Klimenko, Yanqi Qiu, Klaus Schmidt and Maria Shcherbina for useful dis- cussions. The author is supported by A*MIDEX project (No. ANR-11-IDEX- 0001-02), financed by Programme “Investissements d’Avenir” of the Gov- ernment of the French Republic managed by the French National Research Agency (ANR). This work has been supported in part by the Grant MD- 4893.2014.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, by the ANR under the project VALET of the Programme JCJC SIMI 1, and by the RFBR grants 12-01-31284, 12-01-33020, 13-01-12449. Part of this work was done at the Institut Mittag-Leffler of the Royal Swedish Academy of Sciences in Djursholm, Institut Henri Poincare´ in THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 21

Paris and the Institute of Mathematics “Guido Castelnuovo” of the Uni- versity of Rome “La Sapienza”. I am deeply grateful to these institutions for their warm hospitality.

2. CONSTRUCTION AND PROPERTIES OF INFINITE DETERMINANTAL MEASURES 2.1. Preliminary remarks on sigma-ﬁnite measures. Let Y be a Borel ∞ space, and consider a representation Y = Yn of Y as a countable union n=1 of an increasing sequence of subsets Y , Y Y . As before, given a n Sn ⊂ n+1 measure µ on Y and a subset Y 0 Y , we write µ for the restriction of µ ⊂ |Y 0 onto Y 0. Assume that for every n we are given a probability measure Pn on Yn. The following proposition is clear.

B Yn A sigma-ﬁnite measure on Y such that | = n exists Proposition 2.1. B B(Yn) P PN Yn if and only if for any N, n, N > n, we have | = n. The measure is PN (Yn) P B unique up to multiplication by a constant.

Corollary 2.2. If B1, B2 are two sigma-ﬁnite measures on Y such that for B1 Yn all n N we have 0 < B1(Yn) < + , 0 < B2(Yn) < + , and | = ∈ ∞ ∞ B1(Yn) B2 Yn | , then there exists a positive constant C > 0 such that 1 = C 2. B2(Yn) B B 2.2. The unique extension property.

2.2.1. Extension from a subset. Let E be a standard Borel space, let µ be a sigma-ﬁnite measure on E, let L be a closed subspace of L2(E, µ), let Π be the operator of orthogonal projection onto L, and let E0 E be a Borel subset. We shall say that the subspace L has the unique extension⊂ property from E0 if a function ϕ L satisfying χE0 ϕ = 0 must be the zero function and the subspace χ L is∈ closed. In general, if a function ϕ L satisfying E0 ∈ χE0 ϕ = 0 must be the zero function, then the restricted subspace χE0 L still need not be closed: nonetheless, we have the following clear corollary of the open mapping theorem. Proposition 2.3. Assume that the closed subspace L is such that a function

ϕ L satisfying χE0 ϕ = 0 must be the zero function. The subspace χE0 L is closed∈ if and only if there exists ε > 0 such that for any ϕ L we have ∈

(32) χE E0 ϕ (1 ε) ϕ , || \ || ≤ − || || in which case the natural restriction map ϕ χ ϕ is an isomorphism of → E0 Hilbert spaces. If the operator χE E0 Π is compact, then the condition (32) holds. \ 22 ALEXANDER I. BUFETOV

Remark. The condition (32) holds a fortiori if the operator χE E0 Π is \ Hilbert-Schmidt or, equivalently, if the operator χE E0 ΠχE E0 is of trace class. \ \ The following corollaries are immediate. Corollary 2.4. Let g be a bounded nonegative Borel function on E such that (33) inf g(x) > 0. x E0 ∈ If (32) holds then the subspace √gL is closed in L2(E, µ). Remark. The apparently superﬂuous square root is put here to keep notation consistent with the remainder of the paper. Corollary 2.5. Under the assumptions of Proposition 2.3, if (32) holds and a Borel function g : E [0, 1] satisﬁes (33), then the operator Πg of → orthogonal projection onto the subspace √gL is given by the formula g 1 1 (34) Π = √gΠ(1 + (g 1)Π)− √g = √gΠ(1 + (g 1)Π)− Π√g. − − In particular, the operator ΠE0 of orthogonal projection onto the subspace

χE0 L has the form

E0 1 1 (35) Π = χE0 Π(1 χE E0 Π)− χE0 = χE0 Π(1 χE E0 Π)− ΠχE0 . − \ − \ Corollary 2.6. Under the assumptions of Proposition 2.3, if (32) holds, then, for any subset Y E , once the operator χ ΠE0 χ belongs to the ⊂ 0 Y Y trace class, it follows that so does the operator χY ΠχY , and we have trχ ΠE0 χ trχ Πχ Y Y ≥ Y Y E0 Indeed, from (35) it is clear that if the operator χY Π is Hilbert-Schmidt, then the operator χY Π is also Hilbert-Schmidt. The inequality between traces is also immediate from (35).

2.2.2. Examples: the Bessel kernel and the modiﬁed Bessel kernel.

Proposition 2.7. (1) For any ε > 0, the operator J˜s has the unique extension property from the subset (ε, + ); ∞ (2) For any R > 0, the operator J (s) has the unique extension property from the subset (0,R). Proof. The first statement is an immediate corollary of the uncertainty principle for the Hankel transform: a function and its Hankel transform cannot both have support of finite measure [15], [16] (note here that the uncertainty principle is only formulated for s > 1/2 in [15] but the more general uncertainty principle of [16] is directly applicable− also to the case THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 23 s [ 1, 1/2]) and the following estimate, which, by definition, is clearly valid∈ − for any R > 0: R J˜ (y, y)dy < + . s ∞ Z0 The second statement follows from the first by the change of variable y = 4/x. The proposition is proved completely.

2.3. Inductively determinantal measures. Let E be a locally compact complete metric space, and let Conf(E) be the space of configurations on E endowed with the natural Borel structure (see, e.g., [20], [36] ). Given a Borel subset E0 E, we let Conf(E,E0) be the subspace of ⊂ configurations all whose particles lie in E0. Let µ be a σ-finite Borel measure on E. We let E E be a Borel subset 0 ⊂ and assume that for any bounded Borel subset B E E0 we are given a E0 B ⊂ \ closed subspace L L2(E, µ) such that the corresponding projection E0 B S ⊂ operator Π belongs to the space I1,loc(E, µ). We furthermore make the followingS

E0 B E0 B Assumption 1. (1) χBΠ < 1 , χBΠ χB I1(E, µ); (2) (1) (1) (2) k S k E0 SB ∈ E0 B (1) (2) if B B E E0, then χE0 B L = L . ⊂ ⊂ \ S S Proposition 2.8. Under these assumptions,S there exists a σ-finite measure B on Conf(E) such that (1) for B-almost every configuration, only finitely many of its particles may lie in E E0 ; (2) for any bounded\ Borel subset B E E , we have ⊂ \ 0 0 < B(Conf(E; E0 B)) < + and ∪ ∞

B Conf(E; E0 B). | = PΠE0 B . B(Conf(E; E0S B)) S Such a measure will be called an Sinductively determinantal measure. Proposition 2.8 is immediate from Proposition 2.1 . Note that conditions 1 and 2 define our measure uniquely up to multiplication by a constant. We now give a sufficient condition for an inductively determinantal measure to be an actual finite determinantal measure. Proposition 2.9. Consider a family of projections ΠE0 B satisfying the Assumption 1 and the corresponding inductively determinantalS measure B. If there exists R > 0, ε > 0 such that for all bounded Borel subset B E E we have ⊂ \ 0 E0 B (1) χBΠ < 1 ε; k E0S Bk − (2) trχBΠ χB < R. S 24 ALEXANDER I. BUFETOV then there exists a projection operator Π I1,loc(E, µ) onto a closed subspace L L (E, µ) such that ∈ ⊂ 2 E0 B (1) L = χE0 BL for all B; S (2) χE E0 ΠχE E0 S I1(E, µ); \ \ ∈ (3) the measures B and PΠ coincide up to multiplication by a constant.

Proof. By our assumptions, for every bounded Borel subset B E E0 E0 B ⊂ E0 \ B we are given a closed subspace L ∪ , the range of the operator Π , S which has the property of unique extension from E0. The uniform esti- E0 B mate on the norms of the operators χBΠ implies the existence of a E0 B S closed subspace L such that L ∪ = χE0 BL. Now, by our assump- ∪ E0 B tions, the projection operator Π belongs to the space I1,loc(E, µ), S E0 Y whence, for any bounded subset Y E, we have χY Π ∪ χY I1(E, µ), whence, by Corollary 2.6 applied⊂ to the subset E Y , it∈ follows that 0 ∪ χY ΠχY I1(E, µ). It follows that the operator Π of orthogonal projection on L is locally∈ of trace class and therefore induces a unique determinantal probability measure PΠ on Conf(E). Applying Corollary 2.6 again, we have trχE E0 ΠχE E0 R, and the proposition is proved completely \ \ ≤ We now give sufﬁcient conditions for the measure B to be inﬁnite. Proposition 2.10. Make either of the two assumptions:

(1) for any ε > 0, there exists a bounded Borel subset B E E0 such E0 B ⊂ \ that χBΠ > 1 ε; || S || − (2) for any R > 0, there exists a bounded Borel subset B E E0 such E0 B ⊂ \ that trχBΠ χB > R . S Then the measure B is inﬁnite. Proof. Recall that we have

B(Conf(E; E0)) (36) = PΠE0 B (Conf(E; E0)) = B(Conf(E; E0 B)) S E0 B S = det(1 χBΠ χB). − S Under the ﬁrst assumption, it is immediate that the top eigenvalue of the E0 B self-adjoint trace-class operator χBΠ χB exceeds 1 ε, whence det(1 E0 B S − − χBΠ χB) ε. Under the second assumption, write S ≤ E0 B E0 B (37) det(1 χBΠ χB) 6 exp( trχBΠ χB) exp( R). − S − S ≤ − In both cases, the ratio

B(Conf(E; E0)) B(Conf(E; E0 B)) can be made arbitrary small by an appropriateS choice of B, which implies that the measure B is inﬁnite. The proposition is proved. THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 25

2.4. General construction of infinite determinantal measures. By the Macch`ı-Soshnikov Theorem, under some additional assumptions, a determinantal measure can be assigned to an operator of orthogonal projection, or, in other words, to a closed subspace of L2(E, µ). In a similar way, an infinite determinantal measure will be assigned to a subspace H of locally square-integrable functions. Recall that L2,loc(E, µ) is the space of all measurable functions f : E C such that for any bounded subset B E we 2 → ⊂ have f dµ < + . Choosing an exhausting family Bn of bounded sets B | | ∞ (for instance,R balls with fixed centre and of radius tending to infinity), we endow the space L2,loc(E, µ) with a countable family of seminorms which turns it into a complete separable metric space; the topology thus defined does not, of course, depend on the specific choice of the exhausting family. Let H L2,loc(E, µ) be a linear subspace. If E0 E is a Borel subset ⊂ ⊂ E0 such that χE0 H is a closed subspace of L2(E, µ), then we denote by Π the operator of orthogonal projection onto the subspace χE0 H L2(E, µ). We now fix a Borel subset E E; informally, E is the set where⊂ the particles 0 ⊂ 0 accumulate. We impose the following assumption on E0 and H.

Assumption 2. (1) For any bounded Borel set B E, the space χE0 BH ⊂ ∪ is a closed subspace of L2(E, µ); (2) For any bounded Borel set B E E , we have ⊂ \ 0 E0 B E0 B (38) Π ∪ I (E, µ), χ Π ∪ χ I (E, µ); ∈ 1,loc B B ∈ 1 (3) If ϕ H satisfies χ ϕ = 0, then ϕ = 0. ∈ E0 If a subspace H and the subset E have the property that any ϕ H 0 ∈ satisfying χE0 ϕ = 0 must be the zero function, then we shall say that H has the property of unique extension from E0. Theorem 2.11. Let E be a locally compact complete metric space, and let µ be a σ-finite Borel measure on E. If a subspace H L2,loc(E, µ) and a Borel subset E E satisfy Assumption 2, then there exists⊂ a σ-finite Borel 0 ⊂ measure B on Conf(E) such that (1) B-almost every configuration has at most finitely many particles outside of E0; (2) for any bounded Borel (possibly empty) subset B E E we have ⊂ \ 0 0 < B(Conf(E; E0 B)) < + and ∪ ∞

B Conf(E;E0 B) ∪ E B | = PΠ 0∪ . B(Conf(E; E0 B)) ∪ The requirements (1) and (2) determine the measure B uniquely up to multiplication by a positive constant. 26 ALEXANDER I. BUFETOV

We denote B(H,E0) the one-dimensional cone of nonzero inﬁnite determinantal measures induced by H and E0, and, slightly abusing notation, we write B = B(H,E0) for a representative of the cone. Remark. If B is a bounded set, then, by deﬁnition, we have B(H,E ) = B(H,E B). 0 0 ∪

Remark. If E0 E is a Borel subset such that χE0 E is a closed sub- ∪ 0 ⊂ E0 E0 space in L2(E, µ) and the operator Π ∪ of orthogonal projection onto the subspace χE0 E H satisﬁes ∪ 0 E0 E E0 E (39) Π ∪ 0 I (E, µ), χ Π ∪ 0 χ I (E, µ), ∈ 1,loc E0 E0 ∈ 1 then, exhausting E0 by bounded sets, from Theorem 2.11 one easily obtains 0 < B(Conf(E; E0 E0)) < + and ∪ ∞

B Conf(E;E0 E0) | ∪ = E E . PΠ 0∪ 0 B(Conf(E; E0 E0)) ∪ 2.5. Change of variables for infinite determinantal measures. Let F : E E be a homeomorphism. The homeomorphism F induces a homeomorphism→ of the space Conf(E), for which, slightly abusing notation, we keep the same symbol: given X Conf(E), the particles of the configuration F (X) have the form F (x) over∈ all x X. ∈ Assume now that the measures F µ and µ are equivalent, and let B = ∗ B(H,E0) be an infinite determinantal measure. Introduce the subspace dF µ F ∗H = ϕ(F (x)) ∗ , ϕ H . { · s dµ ∈ } From the definitions we now clearly have the following Proposition 2.12. The push-forward of the infinite determinantal measure B = B(H,E0) has the form

F B = B(F ∗H,F (E0)). ∗ 2.6. Example: infinite orthogonal polynomial ensembles. Let ρ be a nonnegative function on R not identically equal to zero. Take N N and ∈ endow the set RN with the measure N (40) (x x )2 ρ(x )dx . i − j i i 1 i,j N i=1 6Y6 Y If for k = 0,..., 2N 2 we have − + ∞ xkρ(x)dx < + , ∞ Z−∞ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 27 then the measure (40) has finite mass and, after normalization, yields a determinantal point process on Conf(R). For a general function ρ, introduce the subspace H(ρ) L2,loc(R, Leb) by the formula ⊂ N 1 H(ρ) = span( ρ(x), x ρ(x), . . . , x − ρ(x)). The measure (40) is an infinitep determinantalp measure,p as is shown by the following immediate Proposition 2.13. Let ρ be a positive continuous function on R. Then for any a, b R, a < b, the measure (40) is an infinite determinantal measure ∈ of the form B(H(ρ), [a, b]). 2.7. Multiplicative functionals of infinite determinantal measures. Our next aim is to show that, under some additional assumptions, an infinite determinantal measure can be represented as a product of a finite determinantal measure and a multiplicative functional.

Proposition 2.14. Let a subspace H L2,loc(E, µ) and a Borel subset ⊂ E0 induce an inﬁnite determinantal measure B = B(H,E0). Let g : E → (0, 1] be a positive Borel function such that √gH is a closed subspace in g L2(E, µ), and let Π be the corresponding projection operator. Assume additionally E0 (1) √1 gΠ √1 g I1(E, µ) ; − g − ∈ (2) χE E0 Π χE E0 I1(E, µ) ; \ \ (3) Πg I (E, µ) ∈ 1,loc Then the multiplicative functional Ψg is B-almost surely positive, B- integrable, and we have

ΨgB = PΠg . Ψg dB Conf(E) R Before starting the proof, we prove some auxiliary propositions. First, we note a simple corollary of unique extension property.

Proposition 2.15. Let H L2,loc(E, µ) have the property of unique exten- ⊂ sion from E0, and let ψ L2,loc(E, µ) be such that χE0 Bψ χE0 BH for any bounded Borel set∈B E E . Then ψ H. ∈ ⊂ \ 0 ∈ S S Proof. Indeed, for any B there exists ψ L (E, µ) such that χ ψ = B ∈ 2,loc E0 B B χE0 Bψ. Take two bounded Borel sets B1 and B2 and note that χE0 ψBS1 = χE0 ψSB2 = χE0 ψ , whence, by the unique extension property, ψB1 = ψB2 . Thus all the functions ψB coincide and also coinside with ψ, which, consequently, belongs to H. 28 ALEXANDER I. BUFETOV

Our next proposition gives a sufﬁcient condition for a subspace of locally square-integrable functions to be a closed subspace in L2.

Proposition 2.16. Let L L2,loc(E, µ) be a subspace such that ⊂ (1) for any bounded Borel set B E E the space χ L is a closed ⊂ \ 0 E0 B subspace of L2(E, µ); S (2) the natural restriction map χ L χ L is an isomorphism of E0 B → E0 Hilbert spaces, and the norm ofS its inverse is bounded above by a positive constant independent of B.

Then L is a closed subspace of L2(E, µ), and the natural restriction map L χ L is an isomorphism of Hilbert spaces. → E0 Proof . If L contained a function with non-integrable square, then for an appropriately chosen of B the inverse of the restriction isomorphism χ L χ L would have an arbitrarily large norm. That L is closed E0 B → E0 followsS from the unique extension property and Proposition 2.15. We now proceed with the proof of Proposition 2.14. First we check that for any bounded Borel B E E we have ⊂ \ 0

E0 B (41) 1 gΠ 1 g I1(E, µ) − S − ∈ Indeed, the deﬁnitionp of an inﬁnitep determinantal measure implies E0 B χBΠ I2(E, µ), S ∈ whence, a fortiori, we have

E0 B 1 gχBΠ I2(E, µ). − S ∈ Now recall that p E0 E0 B E0 B 1 E0 B Π = χE0 Π (1 χBΠ )− Π χE0 . S − S S The relation 1 gΠE0 1 g I (E, µ) − − ∈ 1 therefore implies p p E0 B 1 gχE0 Π χE0 1 g I1(E, µ), − S − ∈ or, equivalently,p p E0 B 1 gχE0 Π I2(E, µ). − S ∈ We coincide that p E0 B 1 gΠ I2(E, µ), − S ∈ or, equivalently, that p E0 B 1 gΠ 1 g I1(E, µ) − S − ∈ as desired. p p THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 29

We next check that the subspace √gHχE0 B is closed in L2(E, µ). But this is immediate from closedness of the subspaceS √gH, the unique extension property from the subset E0, which the subspace √gH has, since so does H, and our assumption g χE E0 Π χE E0 I1(E, µ). \ \ ∈ gχ We now let Π E0 B be the operator of orthogonal projection onto the S subspace √gHχE0 B. It follows from the above that for any bounded Borel set B E E the S ⊂ \ 0 multiplicative functional Ψg is PΠE0 B -almost surely positive and, furthermore, that we have S

ΨgPΠE0 B gχE B S = PΠ 0 , E B Ψg dPΠ 0 S gχ S where Π E0 B is the operator of orthogonal projection onto the closed S R subspace √gχE0 BH. It follows now that for any bounded Borel B E E we have S ⊂ \ 0 Ψgχ E0 B B (42) = gχE B . S PΠ 0 Ψgχ d E0 B B S S It remains to note that (42)R immediately implies the statement of Proposition 2.14, whose proof is thus complete.

2.8. Infinite determinantal measures obtained as finite-rank perturbations of determinantal probability measures. 2.8.1. Construction of finite-rank perturbations. We now consider infinite determinantal measures induced by subspaces H obtained by adding a finite- dimensional subspace V to a closed subspace L L (E, µ). ⊂ 2 Let, therefore, Q I1,loc(E, µ) be the operator of orthogonal projection onto a closed subspace∈ L L (E, µ), let V be a finite-dimensional sub- ⊂ 2 space of L2,loc(E, µ) such that V L2(E, µ) = 0, and set H = L + V . Let E E be a Borel subset. We shall∩ need the following assumption on L, V 0 ⊂ and E0.

Assumption 3. (1) χE E0 QχE E0 I1(E, µ); (2) χ V L (E, µ); \ \ ∈ E0 ⊂ 2 (3) if ϕ V satisﬁes χE0 ϕ χE0 L, then ϕ = 0; (4) if ϕ ∈ L satisﬁes χ ϕ =∈ 0, then ϕ = 0. ∈ E0 Proposition 2.17. If L and E0 satisfy Assumption 2, while L, V and E0 satisfy Assumption 3, then the subspace H = L + V and the subset E0 satisfy Assumption 2. 30 ALEXANDER I. BUFETOV

In particular, for any bounded Borel subset B, the subspace χE0 BL is ∪ closed, as one sees by taking E0 = E B in the following clear 0 ∪ Proposition 2.18. Let Q I1,loc(E, µ) be the operator of orthogonal pro- ∈ jection onto a closed subspace L L (E, µ). Let E0 E be a Borel ∈ 2 ⊂ subset such that χ Qχ I (E, µ) and that for any function ϕ L, the E0 E0 ∈ 1 ∈ equality χE0 ϕ = 0 implies ϕ = 0. Then the subspace χE0 L is closed in L2(E, µ).

The subspace H and the Borel subset E0 therefore define an infinite determinantal measure B = B(H,E0). The measure B(H,E0) is indeed infinite by Proposition 2.10. 2.8.2. Multiplicative functionals of finite-rank perturbations. Proposition 2.14 now has the following immediate

Corollary 2.19. Let L, V and E0 induce an infinite determinantal measure B. Let g : E (0, 1] be a positive measurable function. If → (1) √gV L2(E, µ) ; ⊂ (2) √1 gΠ√1 g I (E, µ) , − − ∈ 1 then the multiplicative functional Ψg is B-almost surely positive and integrable with respect to B, and we have ΨgB = PΠg , Ψg dB g where Π is the operator of orthogonalR projection onto the closed subspace √gL + √gV . 2.9. Example: the infinite Bessel point process. We are now ready to prove Proposition 1.1 on the existence of the infinite Bessel point process B(s), s 1. We first need the following property of the usual Bessel point 6 − process J , s > 1. As before, let L be the range of the projection operator s − s Jes. e e Lemma 2.20. Let s > 1 be arbitrary. Then e − (1) For any R > 0 the subspace χ(R,+ )Ls is closed in L2 (0, + ), Leb , ∞ ∞ and the corresponding projection operator Js,R is locally of trace class; e (2) For any R > 0 we have e P (Conf((0, + ), (R, + ))) > 0, Js ∞ ∞ and e PJ Conf((0,+ ),(R,+ )) s | ∞ ∞ = P . P (Conf((0, + ), (R, + ))) Js,R Js e ∞ ∞ e e THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 31

Proof. First, for any R > 0 we clearly have

R J (x, x) dx < + s ∞ Z0 or, equivalently, e χ J χ I (0, + ), Leb . (0,R) s (0,R) ∈ 1 ∞ The Lemma follows now from the unique extension property of the Bessel e point process. The Lemma is proved completely. (s) Proposition 2.21. We have dimV = ns and for any R > 0 we have

(s) s+2ns χ(R,+ )V χ(R,+ )L = 0. ∞ e ∞ \ Proof. The following argumente has beene suggested by Yanqi Qiu. By deﬁnition of the Bessel kernel, every function lying in Ls+2ns is in fact a restriction onto R+ of a harmonic function deﬁned on the half-plane z : (z) > 0 . The desired claim follows now from the uniqueness theorem{

3. CONVERGENCEOFDETERMINANTALMEASURES. 3.1. Convergence of operators and convergence of measures. The following proposition is well-known.

Proposition 3.1. Assume that the operators Kn I1,loc E, µ , n N, ∈ ∈ K I1,loc E, µ induce determinantal probability measures PKn , n N, ∈ ∈ on Conf(E). If K K in I E, µ as n , then PK n 1,loc PKn PK with respect to the weak→ topology on M (Conf(E))→as ∞n . → ﬁn → ∞ Corollary 3.2. For any s > 1, we have − K˜ (s) J˜ in I ((0, + ), Leb) n → s 1,loc ∞ and

P ˜ (s) PJ˜ in MﬁnConf((0, + )). Kn → s ∞ 32 ALEXANDER I. BUFETOV

Our next aim is to show that, under certain additional assumptions, the convergence above persists under passage to induced processes as well as to ﬁnite-rank perturbations. We proceed to precise statements.

3.2. Convergence of induced processes. Recall that if Π is a projection operator acting on L2(E, µ) and g is a nonnegative bounded measurable function on E such that the operator 1 + (g 1)Π is invertible, then we have set − 1 B˜ (g, Π) = √gΠ(1 + (g 1)Π)− √g. − We now ﬁx g and establish the connection between convergence of the sequence Πn and the corresponding sequence B˜ (g, Πn).

Proposition 3.3. Let Πn, Π I1,loc be orthogonal projection operators, and let g : E [0, 1] be a measurable∈ function such that → 1 gΠ 1 g I1(E, µ), 1 gΠn 1 g I1(E, µ), n N. − − ∈ − − ∈ ∈ Assumep furthermorep that p p (1) Π Π in I (E, µ) as n ; n → 1,loc → ∞ (2) lim tr√1 gΠn√1 g = tr√1 gΠ√1 g; n − − − − (3) the→∞ operator 1 + (g 1)Π is invertible. − Then the operators 1 + (g 1)Πn are also invertible for all suf- ﬁciently large n, and we have−

B˜ (g, Π ) B˜ (g, Π) in I (E, µ) n → 1,loc and, consequently,

P ˜ P ˜ B(g,Πn) → B(g,Π) with respect to the weak topology on M (Conf(E)) as n . fin → ∞ Remark. The second requirement could have been replaced by the requirement that (g 1)Πn converge to (g 1)Π in norm, which is weaker and is what we shall− actually use in the proof− ; nonetheless, in applications it will be more convenient to check the convergence of traces rather than the norm convergence of operators. Proof. The first two requirements and Grumm’s¨ Theorem (see Simon [35]) imply that 1 gΠ 1 gΠ in I (E, µ), − n → − 2 whence, a fortiori,p p (g 1)Π (g 1)Π − n → − THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 33 in norm as n . We now take a bounded Borel subset D E and check that, as n →, ∞ we have ⊂ → ∞ (43) χ B˜ (g, Π )χ χ in I (E, µ). D n D → D 1 Our assumptions directly imply the norm convergence 1 1 (44) (1 + (g 1)Π )− (1 + (g 1)Π)− . − n → − Furthermore, χ Π χ Π as n in the strong operator topology; be- D n → D → ∞ sides, by our assumptions, we have lim trχDΠnχD = trχDΠχD, whence, n →∞ by Grumm’s¨ Theorem, we have χDΠn χDΠ in Hilbert-Schmidt norm, and, a fortiori, in norm. It follows that→ the convergence (43) also takes place in norm. To verify the desired I1 convergence, by Grumm’s¨ Theo- rem again, it suffices to check the relation

(45) lim trχDB˜ (g, Πn)χD = trχDB˜ (g, Π)χD. n →∞ First, if A is a bounded operator, and K1,K2 I2, then one directly veri- ﬁes the inequality ∈

tr(K∗AK ) K A K . 1 2 ≤ || 1||I2 · || || · || 2||I2 It easily follows that the function tr(K1∗AK2) is continuous as long as K1,K2 are operators in I2, and A is a bounded operator. The desired convergence of traces (45) follows from the said continuity since 1 χ B˜ (g, Π)χ = χ g 1Π(1 + (g 1)Π)− Π g 1χ , D D D − − − D and we have the norm convergencep (44) and the I2-convergencep χ Π χ Π. D n → D 3.2.1. Convergence of ﬁnite-rank perturbations. We now proceed to the study of convergence of ﬁnite-rank perturbations of locally trace-class projection operators. Let Ln, L L2(E, µ) be closed subspaces, and let Πn, Π be the corresponding orthogonal⊂ projection operators. Assume we are given (n) non-zero vectors v L2(E, µ), n N, v L2(E, µ), and let Πn, Π be the operators of orthogonal∈ projection∈ onto,∈ respectively, the subspaces (n) Ln + Cv , n N and L Cv . e e ∈ ⊕ Proposition 3.4. Assume

(1) Πn Π in the strong operator topology as n ; (n)→ → ∞ (2) v v in L2(E, µ) as n ; (3) v / L→. → ∞ ∈ Then Πn Π in the strong operator topology as n . If, additionally,→ → ∞ e e Π Π in I (E, µ) as n , n → 1,loc → ∞ 34 ALEXANDER I. BUFETOV then also Π Π in I (E, µ) as n . n → 1,loc → ∞ Let (v, H) stands for the angle between a vector v and a subspace H. ∠ e e Our assumptions imply that there exists α0 > 0 such that (v ,L ) α . ∠ n n ≥ 0 Decompose (n) (n) (n) v = β(n)v1 + v , (n) (n) (n) where v L⊥, v = 1, v L . In this case we have ∈ n k k ∈ n e b Πn = Πn + Pv(n) , e e (n) (n)b where Pv(n) : v v, v v , is the operatore of the orthogonal projection → h (n) i e onto the subspace Cv . e Similarly, decomposee e e v = βv + v with v L⊥, v = 1 , v L, and, again, write ∈ k k ∈ e b Πn = Πn + Pv1 , e e b with Pv(v) = v, v v. e h i e (n) Our assumptions 2 and 3 imply that v v in L2(E, µ). It follows that e → P (n) P in the strong operator topology and also, since our operators v → v e e have one-dimensional range, in I1,loc(E,e µ), whiche implies the proposition. Thee case ofe perturbations of higher rank follows by induction. Let m N (n) (n) (∈n) be arbitrary and assume we are given non-zero vectors v , v , . . . , vm 1 2 ∈ L2(E, µ), n N, v1, v2, . . . , vm L2(E, µ). Let ∈ ∈ (n) (n) Ln = Ln Cv1 Cvm , ⊕ ⊕ · · · ⊕ L = L Cv1 Cvm , e ⊕ ⊕ · · · ⊕ and let Πn, Π be the corresponding projection operators. Applying Propositione 3.4 inductively, we obtain e e Proposition 3.5. Assume

(1) Πn Π in the strong operator topology as n ; (n)→ → ∞ (2) v v in L (E, µ) as n for any i = 1, . . . , m ; i → i 2 → ∞ (3) vk / L Cv1 Cvk 1, k = 1, . . . , m. ∈ ⊕ ⊕ · · · ⊕ − Then Π Π in the strong operator topology as n . If, additionally, n → → ∞ Πn Π in I1,loc(E, µ) as n , e e → → ∞ then also Π Π in I (E, µ) as n , n → 1,loc → ∞ e e THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 35

M (Conf(E)) and, consequently, PΠn PΠ with respect to the weak topology on fin as n . → → ∞ e e 3.3. Application to infinite determinantal measures. Take a sequence (n) (n) B = B(H ,E0) of infinite determinantal measures with H(n) = L(n) + V (n), where L(n) is, as before, the range of a projection operator Π(n) I (E, µ), and V (n) ∈ 1,loc is finite-dimensional. Note that the subset E0 is fixed throughout. Our aim is to give sufficient conditions for convergence of B(n) to a limit measure B = B(H,E0), H = L + V , the subspace L being the range of a projection operator Π I (E, µ). ∈ 1,loc Proposition 3.6. Assume (n) (1) Π Π in I1,loc(E, µ) as n ; → (n) → ∞(n) (n) (2) the subspace V admits a basis v1 , . . . , vm and the subspace V admits a basis v1, . . . , vm such that v(n) v in L (E, µ) as n for all i = 1, . . . , m. i → i 2,loc → ∞ Let g : E [0, 1] be a positive measurable function such that → (1) √1 gΠ(n)√1 g I (E, µ), √1 gΠ√1 g I (E, µ) ; − − ∈ 1 − − ∈ 1 (2) lim tr√1 gΠ(n)√1 g = tr√1 gΠ√1 g ; n − − − − →∞ (n) (3) √gV L2(E, µ), √gV L2(E, µ) ; (n) ⊂ ⊂ (4) √gv √gvi in L2(E, µ) as n for all i = 1, . . . , m. i → → ∞ Then (1) the subspaces √gH(n) and √gH are closed ; (2) the operators Π(g,n) of orthogonal projection onto the subspace √gH(n) and the operator Πg of orthogonal projection onto the subspace √gH satisfy Π(g,n) Πg in I (E, µ) as n . → 1,loc → ∞ Corollary 3.7. In the notation and under the assumptions of Proposition 3.6, we have (n) (1) Ψg L1(Conf(E), B ) for all n, Ψg L1(Conf(E), B); (2) ∈ ∈ (n) ΨgB ΨgB (n) Ψg dB → Ψg dB Conf(E) Conf(E) R R with respect to the weak topology on M (Conf(E)) as n . fin → ∞ 36 ALEXANDER I. BUFETOV

Indeed, the Proposition and the Corollary are immediate from the characterization of multiplicative functionals of infinite determinantal measures given in Proposition 2.14 and Corollary 2.19, the sufficient conditions of convergence of induced processes and finite-rank perturbations given in Propositions 3.3, 3.5, and the characterization of convergence with respect to the weak topology on Mfin(Conf(E)) given in Proposition 3.1.

3.4. Convergence of approximating kernels and the proof of Proposi- tion 1.3. Our next aim is to show that, under certain additional assumptions, if a sequence gn of measurable functions converges to 1, then the gn operators Π considered in Proposition 4.7 converge to Q in I1,loc(E, µ). Given two closed subspaces H1,H2 in L2(E, µ), let α(H1,H2) be the angle between H1 and H2, defined as the infimum of angles between all nonzero vectors in H1 and H2; recall that if one of the subspaces has finite dimension, then the infimum is achieved.

Proposition 3.8. Let L, V , and E0 satisfy Assumption 3, and assume additionally that we have V L2(E, µ) = 0. Let gn : E (0, 1] be a sequence of positive measurable functions∩ such that →

(1) for all n N we have √1 gnQ√1 gn I1(E, µ); ∈ − − ∈ (2) for all n N we have √gnV L2(E, µ); ∈ ⊂ (3) there exists α0 > 0 such that for all n we have (√g H, √g V ) α ; ∠ n n ≥ 0 (4) for any bounded B E we have ⊂ inf gn(x) > 0; lim sup gn(x) 1 = 0. n n N,x E0 B x E0 B | − | ∈ ∈ ∪ →∞ ∈ ∪ Then, as n , we have → ∞ Πgn Q in I (E, µ). → 1,loc Using the second remark after Theorem 2.11, one can extend Proposition 3.8 also to nonnegative functions that admit zero values. Here we restrict ourselves to characteristic functions of the form χE0 B with B bounded, in which case we have the following ∪

Corollary 3.9. Let Bn be an increasing sequence of bounded Borel sets exhausting E E . If there exists α > 0 such that for all n we have \ 0 0

α(χE0 Bn H, χE0 Bn V ) α0, ∪ ∪ ≥ then E0 Bn Π ∪ Q in I (E, µ). → 1,loc THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 37

Informally, Corollary 3.9 means that, as n grows, the induced processes of our determinantal measure on subsets Conf(E; E B ) converge to the 0 ∪ n “unperturbed” determinantal point process PQ. Note that Proposition 1.3 is an immediate corollary of Proposition 3.8 and Corollary 3.9. Proof of Proposition 3.8. We start by showing that, as n , we have → ∞ gnQ Q in norm. Indeed, take ε > 0 and choose a bounded set Bε in such a way→ that ε2 trχE (E0 Bε)QχE (E0 Bε) < . \ \ 4 S S

Since gn 1 uniformly on E0 Bε, we have χE0 Bε (gn 1)Q 0 in norm as n→ . Furthermore, we have − → → ∞ S S ε χE (E0 Bε)Q = χE (E0 Bε)Q I2 < . k \ k k \ k 2 S S Consequently, for n sufﬁciently big, we have:

(gn 1)Q 6 χE0 Bε (gn 1)Q + χE (E0 Bε)Q < ε , k − k k − k k \ k S S and, since ε is arbitrary, it follows that gnQ Q in norm as n . In 1 → → ∞ particular, we have (1 + (gn 1)Q)− 1 in norm as n . Now, since g 1 uniformly− on bounded→ sets, for any→ bounded ∞ Borel n → subset B E, we have χB√gnQ χBQ in I2(E, µ) as n . Consequently,⊂ as n , we have → → ∞ → ∞ 1 χ √g Q(1 + (g 1)Q)− Q√g χ χ Qχ B n n − n B → B B in I (E, µ), and, since B is arbitrary, also Qgn Q in I (E, µ) . 1 → 1,loc We now let Vn be the orthogonal complement of √gnL in √gnL+√gnV , (n) and let P˜ be the operator of orthogonal projection onto Vn. By definition, we have Πgn = Qgn + P˜(n) . To complete the proof, it suffices to check that (n) (n) P˜ 0 in I1,loc(E, µ) as n , to do which, since P˜ are projections onto→ subspaces whose dimension→ ∞ does not exceed that of V , it suffices to show that for any bounded set B we have P˜(n) 0 in strong opera- → tor topology as n . Since the angles between subspaces √gnL and → ∞ √gnV are uniformly bounded from below, it suffices to establish the strong convergence to 0 of the operators P (n) of orthogonal projections onto the subspaces √gnV . Let, therefore, ϕ L2(E, µ) be supported in a bounded ∈ Borel set B; it suffices to show that P (n)ϕ 0 as n . But since V L (E, µ) = 0, for any ε > 0 therek existsk a→ bounded→ set ∞B B such 2 ε ⊃ T 38 ALEXANDER I. BUFETOV

χB ψ 2 k k that for any ψ V we have χ ψ < ε . Now write ∈ k Bε k (46) ΠBε ϕ 2 = ϕ, ΠBε ϕ = k k h i = ϕ, χ ΠBε ϕ ϕ χ ΠBε ϕ h B i 6 k k k B k 6 ϕ ε ΠBε ϕ ε ϕ ΠBε ϕ ε ϕ 2 . 6 k k k k 6 k k k k 6 k k Bε It follows that Π ϕ < ε ϕ and, since g 1 uniformly on B0, k k k k n → also that P (n)ϕ < ε ϕ if n is sufﬁciently large. Since ε is arbitrary, k k k k P (n)ϕ 0 as n , and the Proposition is proved completely. k k → → ∞

4. WEAK COMPACTNESS OF FAMILIES OF DETERMINANTAL MEASURES. 4.1. Configurations and finite measures. In order to prove the vanish- ing of the “Gaussian parameter” and to establish convergence of finite- dimensional approximations on the Pickrell set, we will code configurations by finite measures and determinantal measures by measures on the space of probability measures on the phase space. We proceed to precise definitions. Let f be a nonnegative measurable function on E, set Conf (E) = X : f(x) < , f { ∞} x X X∈ and introduce a map σ : Conf (E) M (E) by the formula f f → fin

σf (X) = f(x)δx. x X X∈ ( where δx stands, of course, for the delta-measure at x). Recall that the intensity ξP of a probability measure P on Conf(E) is a sigma-ﬁnite measure on E deﬁned, for a bounded Borel set B E, by the formula ⊂

ξP(B) = #B(X)dP(X). Conf(Z E)

In particular, for a determinantal measure PK corresponding to an operator K on L2(E, µ) admitting a continuous kernel K(x, y), the intensity is, by deﬁnition, given by the formula

ξPK = K(x, x)µ. By deﬁnition, we have the following

Proposition 4.1. Let f : E [0, + ) be continuous, let P be a probability → ∞ measure on Conf(E). If f L1(E, ξP), then P(Conff (E)) = 1. ∈ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 39

Under the assumptions of Proposition 4.1, the map σf is P-almost surely well-defined, and the measure (σf ) P is a Borel probability measure on the ∗ space Mfin(E), that is, an element of the space Mfin(Mfin(E)). 4.2. Weak compactness and weak convergence in the space of configurations and in the space of finite measures. We start by formulating a tightness criterion for such families of measures. Proposition 4.2. Let f be a nonnegative continuous function on E. Let Pα be a family of Borel probability measures on Conf(E) such that { } (1) f L1(E, ξPα) for all α and ∈

sup fdξPα < + ; α ∞ EZ (2) for any ε > 0 there exists a compact set B E such that ε ⊂ sup fdξPα < ε. α E ZBε \ Then the family (σf ) Pα is tight in Mfin(Mfin(E)). ∗ Remark. The assumptions of Proposition 4.2 can be equivalently re- formulated as follows: the measures (σf ) Pα are all well-defined and the ∗ family fξPα is tight in Mfin(E). Proof of Proposition 4.2. Given ε > 0, our aim is to find a compact set C Mfin(E) such that (σf ) Pα(C) > 1 ε for all α. ⊂ ∗ − Let ϕ : E R be a bounded function. Define a measurable function → intϕ : Mfin(E) R by the formula →

intϕ(η) = ϕdη. ZE Given a Borel subset A E, for brevity we write intA(η) = intχA (η). The following proposition⊂ is immediate from local compactness of the space E and the weak compactness of the space of Borel probability measures on a compact metric space.

Proposition 4.3. Let L > 0, εn > 0, lim εn = 0. Let Kn E be compact n →∞ ⊂ ∞ sets such that Kn = E. The set n=1 S η Mﬁn(E) : intE(η) 6 L, intE Kn (η) 6 εn for all n N { ∈ \ ∈ } is compact in the weak topology on Mﬁn(E). The Prohorov Theorem together with the Chebyshev Inequality now immediately implies 40 ALEXANDER I. BUFETOV

Corollary 4.4. Let L > 0, εn > 0, lim εn = 0. Let Kn E be compact n →∞ ⊂ ∞ sets such that Kn = E. Then the set n=1 S

(47) ν Mfin(Mfin(E)) : intE(η)dν(η) 6 L, ( ∈ MfinZ(E)

intE Kn (η)dν(η) 6 εn for all n N \ ∈ ) MfinZ(E) is compact in the weak topology on Mfin(E). Corollary 4.4 implies Proposition 4.2. First, the total mass of the measures fξPα is uniformly bounded, which, by the Chebyshev inequality, implies, for any ε > 0, the existence of the constant L such that for all α we have (σf ) Pα( η Mfin(E): η(E) L ) > 1 ε. ∗ { ∈ ≤ } − Second, tightness of the family fξPα precisely gives, for any ε > 0, a compact set K E satisfying, for all α, the inequality ε ⊂

intE Kε (η)d(σf ) Pα(η) ε. \ ∗ 6 MﬁnZ(E)

Finally, choosing a sequence εn decaying fast enough and using Corollary 4.4, we conclude the proof of Proposition 4.2. We now give sufficient conditions ensuring that convergence in the space of measures on the space of configurations implies convergence of corresponding measures on the space of finite measures. Proposition 4.5. Let f be a nonnegative continuous function on E. Let Pn, n N, P be Borel probability measures on Conf(E) such that ∈ (1) Pn P with respect to the weak topology on Mfin(Conf(E)) as n → ; → ∞ (2) f L1(E, ξPn) for all n N; ∈ ∈ (3) the family fξPn is a tight family of finite Borel measures on E. Then P(Conff (E)) = 1 and the measures (σf ) Pn converge to (σf ) P weakly in M (M (E)) as n . ∗ ∗ fin fin → ∞ Proposition 4.5 easily follows from Proposition 4.2. First, we restrict ourselves to the open subset x E : f(x) > 0 which itself is a complete separable metric space with{ respect∈ to the induced} topology. Next observe that the total mass of the measures fξPα is uniformly bounded, which, by THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 41 the Chebyshev inequality, implies, for any ε > 0, the existence of the constant L such that for all n we have

Pn( X Conf(E): f(X) L ) > 1 ε. { ∈ ≤ } − x X X∈ Since the measures Pn converge to P weakly in Mfin(Conf(E)) and the set X Conf(E): f(X) L is closed in Conf(E), it follows that { ∈ x X ≤ } ∈ P P( X Conf(E): f(X) L ) > 1 ε, { ∈ ≤ } − x X X∈ and, consequently, that P(Conff (E)) = 1, and the measure (σf ) P is well- defined. ∗ The family (σf ) Pn is tight and must have a weak accumulation point P0. ∗ Using the weak convergence Pn P in Mfin(Conf(E)), we now show → that the finite-dimensional distributions of P0 coincide with those of (σf ) P. Here we use the assumption that our function f is positive and, conse-∗ quently, bounded away from zero on every bounded subset of our locally compact space E. Let ϕ1, . . . , ϕl : E R be continuous functions with disjoint compact supports. By definition,→ the joint distribution of the random variables intϕ1 ,..., intϕl with respect to (σf ) Pn coincides with the ∗ joint distribution of the random variables intϕ1/f ,..., intϕl/f with respect to Pn. As n , this joint distribution converges to the joint distribu- → ∞ tion of intϕ1/f ,..., intϕl/f with respect to P which on the one hand, coincides with the the joint distribution of the random variables intϕ1 ,..., intϕl with respect to (σf ) P and, on the other hand, also coincides with the ∗ joint distribution of the random variables intϕ1 ,..., intϕl with respect to P0. Finite-dimensional distributions determine a measure uniquely, whence P0 = (σf ) P, and the proof is complete. ∗ 4.3. Applications to determinantal point processes. Let f be a nonnegative continuous function on E. If an operator K I (E, µ) induces a ∈ 1,loc determinantal measure PK and satisfies fK I1(E, µ), then ∈ (48) PK (Conff (E)) = 1. If, additionally, K is assumed to be self-adjoint, then the weaker requirement √fK√f I1(E, µ) also implies (48). In this special∈ case, a sufficient condition for tightness takes the following form. Proposition 4.6. Let f be a bounded nonnegative continuous function on E. Let Kα I1,loc(E, µ) be a family of self-adjoint positive contractions such that ∈ sup tr fKα f < + α ∞ p p 42 ALEXANDER I. BUFETOV and such that for any ε > 0 there exists a bounded set B E such that ε ⊂

sup trχE Bε fKα fχE Bε < ε. α \ \ p p Then the family of measures (σf ) PKα , is weakly precompact in Mﬁn(Mﬁn(E)). { ∗ } 4.4. Induced processes corresponding to functions assuming values in [0, 1]. Let g : E [0, 1] be a nonnegative Borel function, and, as before, → let Π I (E, µ) be an orthogonal projection operator with range L ∈ 1,loc inducing a determinantal measure PΠ on Conf(E). Since the values of g do not exceed 1, the multiplicative functional Ψg is automatically integrable.

Proposition 4.7. If √1 gΠ√1 g I1(E, µ) and (1 g)Π < 1, then − − ∈ || − ||

(1) Ψg is positive on a set of positive measure; (2) the subspace √gL is closed, and the operator Πg of orthogonal projection onto the subspace √gL is locally of trace class; (3) we have ΨgPΠ (49) PΠg = . Ψg dPΠ Conf(E) R Remark. Since the operator √1 gΠ is, by assumption, Hilbert-Schmidt, and the values of g do not exceed 1,− the condition (1 g)Π < 1 is equivalent to the condition √1 gΠ < 1 and both are|| equivalent− || to the nonex- istence of a function ||Φ L− supported|| on the set x E : g(x) = 0 . In particular, if the function∈g is strictly positive, the condition{ ∈ is automatically} verified. Proposition 4.6 now implies Corollary 4.8. Let f be a bounded nonnegative continuous function on E. Under the assumptions of Proposition 4.7, if tr fΠ f < + , ∞ then also p p tr fΠg f < + . ∞ Proof: Equivalently, we mustp provep that if the operator √fΠ is Hilbert- Schmidt, then the operator √fΠg is also Hilbert-Schmidt. Since Πg = 1 √gΠ(1 + (g 1)Π)− √g, the statement is immediate from the fact that Hilbert-Schmidt− operators form an ideal. 4.5. Tightness for families of induced processes. We now give a sufficient condition for the tightness of families of measures of the form Πg for fixed g. This condition will subsequently be used for establishing convergence of determinantal measures obtained as products of infinite determinantal measures and multiplicative functionals. THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 43

Let Π I (E, µ) be a family of orthogonal projection operators in α ∈ 1,loc L2(E, µ). Let Lα be the range of Πα. Let g : E [0, 1] be a Borel function such that for each α the assumptions of Proposition→ 4.7 are satisﬁed and g g thus the operators Πα and the corresponding determinantal measures PΠα are well-deﬁned for all α. Furthermore, let f be a nonnegative function on E such that such that for all α we have

(50) sup tr fΠα f < + α ∞ p p and such that for any ε > 0 there exists a bounded Borel set Bε E such that ⊂

(51) sup trχE Bε fΠα fχE Bε < ε. α \ \ (in other words, f is such that allp the assumptionsp of Proposition 4.6 are g satisfied for all α). It follows from Corollary 4.8 that the measures (σf ) PΠα are also well-defined for all α. ∗ Sufficient conditions for tightness of this family of operators are given in the following Proposition 4.9. In addition to the requirements, for all α, of Proposition 4.6 and Proposition 4.7, make the assumption

(52) inf 1 (1 g)Πα > 0. α − || − ||

Then the family of measures (σf ) PΠg is weakly precompact in Mfin(Mfin(E)). { ∗ α } Proof. The requirement (52) implies that the norms of the operators 1 (1 + (g 1)Π )− − α g 1 are uniformly bounded in α. Recalling that Πα = √gΠα(1+(g 1)Πα)− √g, we obtain that (50) implies − g (53) sup tr fΠα f < + , α ∞ while (51) implies p p g (54) sup trχE Bε fΠα fχE Bε < ε. α \ \ Proposition 4.9 is now immediatep fromp Proposition 4.6. 4.6. Tightness of families of finite-rank deformations. We next remark that, under certain additional assumptions, tightness is preserved by taking finite-dimensional deformations of determinantal processes. As before, we let Π I (E, µ) be a family of orthogonal projection α ∈ 1,loc operators in L2(E, µ). Let Lα be the range of Πα. Let v(α) L2(E, µ) be v v ∈ orthogonal to Lα, let Lα = Lα Cv(α), and let Πα be the corresponding ⊕ 44 ALEXANDER I. BUFETOV orthogonal projection operator. By the Macch`ı-Soshnikov theorem, the op- v v erator Πα induces a determinantal measure PΠα on Conf(E). As above, we require that all the assumptions of Proposition 4.6 be satisfied for the family Πα. The following Corollary is immediate from Proposition 4.6. Proposition 4.10. Assume additionally that the family of measures f v(α) 2µ | | is precompact in Mfin(E). Then the family of measures (σf ) PΠv , is { ∗ α } weakly precompact in Mfin(Mfin(E)). This proposition can be extended to perturbations of higher rank. The α assumption of orthogonality of v to Lα is too restrictive and can be weak- ened to an assumption that the angle between the vector and the subspace is bounded from below: indeed, in that case we can orthogonalize and apply Proposition 4.10. We thus take m N and assume that, in addition to the ∈ family of Πα of locally trace-class projection operators considered above, (1) (m) for every α we are given vectors vα , . . . , vα of unit length, linearly in- v,m (1) (m) dependent and independent from Lα. Set Lα = Lα Cvα Cvα , v,m ⊕ ⊕ and let Πα be the corresponding projection operator. By the Macch`ı- v,m Soshnikov theorem, the operator Πα induces a determinantal measure v,m PΠα on Conf(E). As above, we require that all the assumptions of Propo- sition 4.6 be satisfied for the family Πα.

(k) 2 Proposition 4.11. Assume additionally that the family of measures f vα µ, | | over all α and k, is precompact in Mfin(E) and that there exists δ > 0 such that for any k = 1, . . . , m and all α we have (k) (1) (k 1) (vα ,Lα Cvα Cvα − ) δ. ∠ ⊕ ⊕ · · · ⊕ ≥ Then the family of measures (σf ) PΠv,m , is weakly precompact in Mfin(Mfin(E)). { ∗ α } The proof proceeds by induction on m. For m = 1, it suffices to apply Proposition 4.10 to the vector obtained by taking the orthogonal pro- (1) jection of vα onto the orthogonal complement of L. For the induction step, similarly, we apply Proposition 4.10 to the vector obtained by tak- (m) ing the orthogonal projection of vα onto the orthogonal complement of (1) (m 1) Lα Cvα Cvα − ). The proposition is proved completely. ⊕ ⊕ · · · ⊕ 4.7. Convergence of finite-rank perturbations. A sufficient condition for weak convergence of determinatal measures considered as elements of the space Mfin(Mfin(E)) can be formulated as follows. Proposition 4.12. Let f be a nonnegative continuous function on E. Let K ,K I be self-adjoint positive contractions such that K K in n ∈ 1,loc n → I (E, µ) as n . Assume additionally that 1,loc → ∞ (55) fK f fK f in I (E, µ) n → 1 p p p p THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 45 as n . Then → ∞ (σf ) PKn (σf ) PK ∗ → ∗ weakly in M(M (E)) as n . fin → ∞ Combining Proposition 4.12 with, on the one hand, Propositions 4.9, 4.11 and, on the other hand, Propositions 3.3, 3.5 and 3.6, we arrive at the following Proposition 4.13. (1) In the notation and under the assumptions of Propo- sition 3.3, additionally require (55) to hold for Πn. Then we have fB˜ (g, Π ) f fB˜ (g, Π) f n → in I1(E, µp), and, consequently,p p p

P ˜ P ˜ B(g,Πn) → B(g,Π) with respect to the weak topology on Mfin(Mfin(E)) as n . (2) In the notation and under the assumptions of Proposition 3.5→, ∞ additionally require (55) to hold for Πn. Then we have fΠ f fΠ f in I (E, µ) as n , n → 1 → ∞ and, consequently,p p Pp pP with respect to the weak topology on e Πn →e Π Mfin(Mfin(E)) as n ; (3) In the notation and under→e ∞ thee assumptions of Proposition 3.6, additionally require (55) to hold for Πn. Then we have fΠ(g,n) f fΠg f in I (E, µ) as n . → 1 → ∞ and,p consequently,p p p

(n) ΨgB ΨgB (n) Ψg dB → Ψg dB

with respect to theR weak topologyR on Mﬁn(Mﬁn(E)) as n . → ∞

5. WEAK CONVERGENCE OF RESCALED RADIAL PARTS OF PICKRELL MEASURES . 5.1. The case s > 1: ﬁnite Pickrell measures. − 5.1.1. Determinantal representation of the radial parts of ﬁnite Pickrell measures. We go back to radial parts of Pickrell measures and start with (s) the case s > 1 . Recall that Pn stand for the Jacobi polynomials corresponding to the− weight (1 u)s on the interval [ 1, 1]. − − 46 ALEXANDER I. BUFETOV

We start by giving a determinantal representation for the radial part of ﬁnite Pickrell measures: in other words, we simply rewrite the formula (6) in the coordinates λ1, . . . , λn. Set

ˆ (s) n(n + s) 1 (56) Kn (λ1, λ2) = s/2 s/2 2n + s (1 + λ1) (1 + λ2) ×

(s) λ1 1 (s) λ2 1 (s) λ2 1 (s) λ1 1 Pn ( λ −+1 )Pn 1( λ +1− ) Pn ( λ +1− )Pn 1( λ −+1 ) 1 − 2 − 2 − 1 . × λ λ 1 − 2 (s) The kernel Kˆn is the image of the Christoffel-Darboux kernel for the Ja- λi 1 cobi ensemble under the change of variable ui = − . Another representa- λi+1 (s) tion for the kernel Kˆn is

ˆ (s) 1 (57) Kn (λ1, λ2) = s/2+1 s/2+1 (1 + λ1) (1 + λ2) × n 1 − (s) λ1 1 (s) λ2 1 (2l + s + 1)Pl − Pl − . × λ1 + 1 · λ2 + 1 Xl=0 (s) The kernel Kˆn is by deﬁnition the kernel of the operator of orthogonal projection in L ((0, + ), Leb) onto the subspace 2 ∞ ˆ(s,n) 1 (s) λ1 1 (58) L = span( s/2+1 Pl ( − ), l = 0, . . . , n 1) = (λ + 1) λ1 + 1 −

1 λ1 1 l = span( s/2+1 ( − ) , l = 0, . . . , n 1). (λ + 1) λ1 + 1 − Proposition 1.17 implies the following determinantal representation the radial part of the Pickrell measure. n (s) 1 ˆ (s) Proposition 5.1. For s > 1, we have (radn) µn = n! det Kn (λi, λj) dλi. − ∗ i=1 Q 5.1.2. Scaling. For β > 0, let homβ : (0, + ) (0, + ) be the homo- thety map that sends x to βx ; we keep the∞ same→ symbol∞ for the induced scaling transformation of Conf((0, + )). We now give an explicit determinantal∞ representation for the measure

(s) (59) (conf homn2 radn) µn , ◦ ◦ ∗ the push-forward to the space of conﬁgurations of the rescaled radial part of (s) the Pickrell measure µn . Consider the rescaled Christoffel-Darboux kernel (s) 2 ˜ (s) 2 2 (60) Kn = n Kn (n λ1, n λ2) THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 47 of orthogonal projection onto the rescaled subspace

2 (s,n) 1 (s) n λ1 1 (61) L = span( 2 s/2+1 Pl ( 2 − )) = (n λ + 1) n λ1 + 1 2 1 n λ1 1 l = span( 2 s/2+1 ( 2 − ) , l = 0, . . . , n 1). (n λ + 1) n λ1 + 1 −

(s) The kernel Kn induces a detrminantal process P (s) on the space Conf((0, + )). Kn ∞ Proposition 5.2. For s > 1, we have − n (s) 1 (s) (homn2 radn) µn = det Kn (λi, λj) dλi. ◦ ∗ n! i=1 Y (s) Equivalently, (conf homn2 radn) µn = P (s) . ◦ ◦ ∗ Kn 5.1.3. Scaling limit. The scaling limit for radial parts of ﬁnite Pickrell measures is a variant of the well-known result of Tracy and Widom [38] claim- ing that the scaling limit of Jacobi orthogonal polynomial ensembles is the Bessel point process.

(s) Proposition 5.3. For any s > 1, as n , the kernel Kn converges to the kernel J (s) uniformly in the− totality of→ variables ∞ on compact subsets of (s) (s) (0, + ) (0, + ). We therefore have Kn J in I ((0, + ), Leb) ∞ × ∞ → 1,loc ∞ and P (s) PJ(s) in MﬁnConf((0, + )). Kn → ∞ Proof. This is an immediate corollary of the classical Heine-Mehler asymptotics for Jacobi polynomials, see, e.g., Szego¨ [37].

5.2. The case s 1: infinite Pickrell measures. ≤ − 5.2.1. Representation of radial parts of infinite Pickrell measures as infinite determinantal measures. Our first aim is to show that for s 1, the measure (31) is an infinite determinantal measure. Similarly to≤ the − definitions given in the Introduction, set 1 1 λ 1 (62) Vˆ (s,n) = span( , ( − ),..., (λ + 1)s/2+1 (λ + 1)s/2+1 λ + 1

1 (s+2ns 1) λ 1 − ..., Pn ns ( − )). (λ + 1)s/2+1 − λ + 1

(s,n) (s,n) (s+2ns,n ns) (63) Hˆ = Vˆ Lˆ − . ⊕ 48 ALEXANDER I. BUFETOV

Consider now the rescaled subspaces 1 1 n2λ 1 (64) V (s,n) = span( , ( − ),..., (n2λ + 1)s/2+1 (n2λ + 1)s/2+1 n2λ + 1 2 1 (s+2ns 1) n λ 1 − ..., Pn ns ( − )). (n2λ + 1)s/2+1 − n2λ + 1

(s,n) (s,n) (s+2ns,n ns) (65) H = V L − . ⊕ Proposition 5.4. Let s 1 , and let R > 0 be arbitrary. The radial part of the Pickrell measure≤ − is then an infinite determinantal measure cor- (s,n) responding to the subspace H = Hˆ and the subset E0 = (0,R): (s) ˆ (s,n) (radn) µn = B(H , (0,R)). ∗ For the rescaled radial part, we have (n) (s) (s) (s,n) conf r (µ = (conf homn2 radn) µn = B(H , (0,R)). ∗ ◦ ◦ ∗ 5.3. The modified Bessel point process as the scaling limit of the radial parts of infinite Pickrell measures: formulation of Proposition 5.5. Denote B(s,n) = B(H(s,n), (0,R)). We now apply the formalism of the previous sections to describe the limit transition of the measures B(s,n) to B(s): namely, we multiply our sequence of infinite measures by a conver- gent multiplicative functional and establish the convergence of the resulting sequence of determinantal probability measures. It will be convenient to take β > 0 and set gβ(x) = exp( βx), while for f it will be convenient to take the function f(x) = min(x, 1)−. Set, therefore, L(n,s,β) = exp( βx/2)H(s,n). − It is clear by definition that L(n,s,β) is a closed subspace of L ((0, + ), Leb); 2 ∞ let Π(n,s,β) be the corresponding orthogonal projection operator. Recall also from (15), (16) the operator Π(s,β) of orthogonal projection onto the subspace L(s,β) = exp( βx/2)H(s). − (s) Proposition 5.5. (1) For all β > 0 we have Ψgβ L1(Conf(0, + ), B ) ∈ ∞(s,n) and, for all n > s+1 we also have Ψgβ L1(Conf(0, + ), B ); (2) we have − ∈ ∞ (s,n) (s) Ψgβ B Ψgβ B (n,s,β) (s,β) (s,n) = PΠ ; (s) = PΠ ; Ψgβ dB Ψgβ dB (n,s,β) s,β (3) we haveR Π Π in RI1,loc((0, + ), Leb)) as n → ∞ → and, consequently, PΠ(n,s,β) PΠ(s,β) as n weakly in ∞M (Conf (0, + ) ); → → ∞ fin ∞ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 49

(4) for f(x) = min(x, 1) we have fΠ(n,s,β) f, fΠs,β f I ((0, + ), Leb)); ∈ 1 ∞ fΠ(n,s,βp) f pfΠs,βp f inpI ((0, + ), Leb)) as n . → 1 ∞ → ∞ p and, consequently,p p p

(σf ) PΠ(g,s,n) (σf ) PΠ(g,s) ∗ → ∗ as n weakly in M (M (0, + ) ). → ∞ fin fin ∞ The proof of Proposition 5.5 will occupy the remainder of this section. 5.4. Proof of Proposition 5.5. 5.4.1. Proof of the first three claims. For s > 1, write − L(n,s,β) = exp( βx/2)L(s,n),L(s,β) = exp( βx/2)L(s) − Jac − and keep the notation Π(n,s,β) , Π(s,β) for the corresponding orthogonal projection operators. For s > 1, using the Proposition 3.3 on the convergence of induced processes, we clearly− have

Ψgβ (s) β (s) PKn Ψg PJ = PΠ(n,s,β) ; = PΠ(s,β) , Ψgβ d (s) Ψgβ d (s) PKn PJ as well as theR convergence R Π(n,s,β) Πs,β in I ((0, + ), Leb)) as n . → 1,loc ∞ → ∞ If xn x as n , then, of course, for any α R we have → → ∞ ∈ 1 2 α α lim (n xn + 1) = x , n 2α →∞ n and, by the Heine-Mehler classical asymptotics, for any α > 1, we also have − 2 2 α/2+1 (α) n xn 1 Jα(2/√x) lim 1(n x + 1) Pn ( − ) = . n 2 →∞ n xn + 1 √x Applying the change of variables y = 4/x to Proposition 2.21, we see 2 Js+2ns 1( ) s/2 1 − √x that for any s 1, and any R > 0 the functions x− − χ , ..., χ 6 − (0,R) √x (0,R) are linearly independent and, furthermore, are independent from the space s+2ns χ(0,R)L . It follows, of course, that the functions 2 Js+2ns 1( ) βx/2 s/2 1 βx/2 − √x (66) e− x− − , . . . , e− √x

βx/2 s+2ns are also linearly independent and independent from the space e− L . The ﬁrst three claims of Proposition 5.5 follow now from its abstract coun- terparts established in the previous subsections: the ﬁrst and the second 50 ALEXANDER I. BUFETOV claim follow from Corollary 2.19, while the third claim, from Proposition 3.6. We proceed to the proof of the fourth and last claim of Proposition 5.5.

5.4.2. The asymptotics J (s) at 0 and at . We shall need the asymptotics of the modiﬁed Bessel kernel J (s) at 0 and∞ at . ∞ We start with a simple estimate for the usual Bessel kernel J˜s. Proposition 5.6. For any s > 1 and any R > 0 we have − + ∞J˜ (y, y) (67) s dy < + . y ∞ ZR Proof. Write + 1 1 + ∞ ∞ 2 1 (Js(√y)) (J (√ty))2dtdy = dt dy = y s y ZR Z0 Z0 tRZ

+ R + ∞ 2 ∞ 2 y Js(√y) Js(√y)) = min , 1 dy = 1/R J (√y))2dx + dy. R · y s y Z0 Z0 ZR It is immediate from the asymptotics of the Bessel functions at zero and at infinity that both integrals converge, and the proposition is proved. Effectu- ating the charge of variable y = 4/x, we arrive at the following Proposition 5.7. For any s > 1 and any ε > 0 there exists δ > 0 such δ (s) − that 0 xJ (x, x)dx < ε. R R 2 Next note that (Js(√y)) dy < + since for a fixed s > 1 and 0 ∞ − 2 s all sufficiently smallR y > 0 we have (Js(√y)) = O(y ). Now, write R 1 R R 2 J˜s(y, y)dy = Js(√ty)dydt (R+1) (Js(√y)) dy < + , whence 0 0 0 ≤ 0 ∞ R R R R R for any R > 0 we have J˜s(y, y)dy < . Making the change of variables 0 ∞ y = 4/x, we obtain R

Proposition 5.8. For any R > 0 we have ∞ J (s)(x, x)dx < . R ∞ 5.4.3. Uniform in n asymptotics at inﬁnityR for the kernels K(n,s). We turn to the uniform asymptotic at inﬁnity for the kernels K(n,s) and the limit kernel J (s). This uniform asymptotics is needed to establish the last claim of Proposition 5.5. THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 51

Proposition 5.9. For any s > 1 and any ε > 0 there exists R > 0 such + − ∞ (n,s) that supn N K (x, x)dx < ε. ∈ R R Proof. If s > 0 then the classical inequalities for Borel functions and Jacobi polynomials (see e.g. Szego¨ [37]) imply the existence of a constant (n,s) C C > 0 such that for all x > 1 we have: sup K (x, x) < x2 . The remain- n N ∈ ing case s ( 1, 0] is immediate from the fact that the kernels K(n,s) are ∈ − (n 1,s+2) rank-one perturbations of the kernels K − .

5.4.4. Uniform in n asymptotics at zero for the kernels K(n,s) and completion of the proof of Proposition 5.5. We next turn to the uniform asymptotics at zero for the kernels K(n,s) and the limit kernel J (s). Again, this uniform asymptotics is needed to establish the last claim of Proposition 5.5.

Proposition 5.10. For any ε > 0 there exists δ > 0 such that for all n N we have ∈ δ (68) xK(n,s)(x, x)dx < ε. Z0 Proof. Going back to the u-variable, we prove that for any ε > 0 there exists R > 0, n0 N, such that for all n > n0 we have ∈ 1 R/n2 1 − 1 + u (69) K˜ (s)(u, u)du < ε. n2 1 u n Z1 − − 1+u First note that the function 1 u is bounded above on [ 1, 0], and therefore − − 0 1 1 1 + u 2 2 K˜ (s)(u, u)du K˜ (s)(u, u)du = . n2 1 u n ≤ n n n Z1 − Z1 − − 1 R/n2 1 − 1+u ˜ (s) We proceed to estimating n2 1 u Kn (u, u)du 0 − Fix κ > 0 (the precise choiceR of κ will be described later). Write

(70) K˜ (s)(u, u) = ( (2l + s + 1)(P (s)(u))2)(1 u)s+ n l − l κn X≤ + ( (2l + s + 1)(P (s))2)(1 u)s. l − l>κnX 52 ALEXANDER I. BUFETOV

We start by estimating

1 R/n2 1 − 1 + u (71) (2l + s + 1)(P (s))2(1 u)sdu. n2 1 u l − l κn ≤ 1 Z1/l2 − X − Using the trivial estimate

(s) 2 max Pl (u) = O(l ) u [ 1,1] | | ∈ − we arrive, for the integral (71), at the upper bound

1 c − n2 1 2s+1 s 1 (72) const l (1 u) − du · n2 · − l κn ≤ 1 Z1 X − l2 We now consider three cases: s > 0, s = 0, and 1 < s < 0. The First Case. If s > 0, then the integral (72) is− estimated above by the expression 1 1 const l2s+1 const κ2. · n2 · l2s ≤ · l κn X≤ The Second Case. If s = 0, then the integral (72) is estimated above by the expression 1 n const l log( ) const κ2. · n2 · l ≤ · l κn X≤ The Third Case. Finally, if 1 < s < 0, then we arrive, for the integral (72), at the upper bound −

1 2s+1 s 2s s 2+2s const ( l ) R n− const R κ · n2 · ≤ · l κn X≤ Note that in this case, the upper bound decreases as R grows. Note that in all three cases the contribution of the integral (72) can be made arbitrarily small by choosing κ sufﬁciently small. We next estimate

1 1 − l2 1 1 + u (73) (sl + s + 1)(P (s)(u))2(1 u)sdu n2 1 u l − l κn X≤ Z0 − Here we use the estimate (7.32.5) in Szego¨ [37] that gives

s 1 (1 u)− 2 − 4 P (s)(u) const − | l | ≤ √n THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 53

1 as long as u [0, 1 l2 ] and arrive, for the integral (73), at the upper bound 1 ∈ 2 − 3 const n3 l const κ , which, again, can be made arbitrarily small · l κn ≤ · ≤ as soon as κPis chosen sufﬁciently small. It remains to estimate the integral

1 R − n2 1 1 + u (74) (2l + s + 1)(P (s))2(1 u)sdu n2 1 u · l − κn l

1 R − n2 const 3 const (1 u)− 2 du n3 − ≤ √ κn l

6. CONVERGENCEOFAPPROXIMATINGMEASURESONTHE PICKRELL SETANDPROOFOF PROPOSITIONS 1.15, 1.16. 6.1. Proof of Proposition 1.15. Proposition 1.15 easily follows from what has already been established. Recall that we have a natural forgetting map conf : Ω Conf(0, + ) that assigns to ω = (γ, x), x = (x , . . . , x ,... ), P → ∞ 1 n the configuration ω(x) = (x1, . . . , xn,... ). By definition, the map conf is r(n)(µ(s)-almost surely bijective. The characterization of the measure conf r(n)(µ(s)) as an infinite determinantal measure given by Proposition 5.4 and∗ the first statement of Proposition 5.5 now imply Proposition 1.15. 6.2. Proof of Proposition 1.16. Recall that, by definition, we have (s,n,β) conf ν = PΠ(s,n,β) . ∗ Recall that Proposition 5.5 implies that, for any s R, β > 0, as n we ∈ → ∞ have PΠ(s,n,β) PΠ(s,β) in Mfin(Conf((0, + ))) and, furthermore, setting → ∞ 54 ALEXANDER I. BUFETOV f(x) = min(x, 1), also the weak convergence (σf ) PΠ(s,n,β) (σf ) PΠ(s,β) ∗ → ∗ in Mfin(Mfin((0, + ))). We now need to pass from weak convergence of probability measures∞ on the space of configurations established in Propo- sition 5.5 to the weak convergence of probability measures on the Pickrell set. We have a natural map s:Ω M (0, + ) defined by the for- P → fin ∞ ∞ mula s(ω) = min(xi(ω), 1)δxi(ω) . The map s is bijective in restriction i=1 0 to the subset ΩPdefined, we recall, as the subset of ω = (γ, x) ΩP such P ∈ that γ = xi(ω). Remark. The function min(x, 1) is chosen only for concreteness: any other positiveP bounded function on (0, + ) coinciding with x on some interval (0, ε) and bounded away from zero∞ on its complement, could have been chosen instead. Consider the set ∞ sΩ = η M (0, + ) : η = min(x , 1)δ for some x > 0 . P { ∈ fin ∞ i xi i } i=1 X The set sΩ is clearly closed in M (0, + ) . Any measure η from the P fin ∞ set sΩ admits a unique representation η = sω for a unique ω Ω0 . Con- P ∈ P sequently, to any finite Borel measure P Mfin(Mfin((0, + ))) supported ∈ ∞ on the set sΩP there corresponds a unique measure pP on ΩP such that 0 s pP = P and pP(ΩP ΩP ) = 0. ∗ \ 6.3. Weak convergence in Mfin(Mfin((0, + ))) and in Mfin(ΩP ). ∞ Proposition 6.1. Let νn, ν Mfin(Mfin((0, + ))) be supported on the set ∈ ∞ sΩP and assume that νn ν weakly in Mfin(Mfin((0, + ))) as n . Then pν pν weakly in→M (Ω ) as n . ∞ → ∞ n → fin P → ∞ ∞ The map s is, of course, not continuous, since the function ω min(xi(ω), 1) → i=1 is not continuous on the Pickrell set. Nonetheless, we have the followingP relation between tightness of measures on Ω and on M (0, + ) . P fin ∞ Lemma 6.2. Let Pα Mfin(Mfin((0, + ))) be a tight family of measures ∈ ∞ supported on the set sΩP . Then the family pPα is also tight. Proof. Take R > 0 and consider the subset ∞ Ω (R) = ω Ω : γ(ω) R, min(x (ω), 1) R . P { ∈ P ≤ i 6 } i=1 X The subset ΩP (R) is compact in ΩP , and any compact subset of sΩP is in fact a subset of sΩP (R) for a sufficiently large R. Consequently, for any ε > 0 one can find a sufficiently large R in such a way that Pα(s(ΩP (R))) > THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 55

0 1 ε for all α. Since all measures Pα are supported on ΩP , it follows that − pPα(ΩP (R)) > 1 ε for all α , and the desired tightness is established. −

Corollary 6.3. Let Pn Mfin(Mfin (0, + ) ), n N, P Mfin(Mfin (0, + ) ) be finite Borel measures.∈ Assume ∞ ∈ ∈ ∞ (1) the measures Pn are supported on the set sΩP for all n N; ∈ (2) Pn P converge weakly in Mfin(Mfin (0, + ) ) as n . → ∞ → ∞ Then the measure P is also supported on the setsΩP andpPn pP weakly in M (Ω ) as n . → fin P → ∞

Proof. The measure P is of course supported on the set sΩP , since the set sΩP is closed. The desired weak convergence in Mfin(ΩP ) is now established in three steps. The First Step: The Family pPn is Tight . The family pPn is tight by Lemma 6.2 and, after passage to a subsequence, weakly converges to a measure P0 Mfin(ΩP ). ∈ The Second Step: Finite-Dimensional Distributions Converge. Let l N, ∈ let ϕl : (0, + ) R be continuous compactly supported functions, set ∞ → ϕl(x) = min(x, 1)ψl(x), take t1, . . . , tl R and observe that, by definition, l ∈ l ∞ for any ω ΩP we have exp(i tk( ϕk(xi(ω)))) = exp(i tkintψk (sω)) ∈ k=1 r=1 k=1 and consequently P P P

l ∞ (75) exp(i tk( ϕk(xi(ω)))) dP0(ω) = Z k=1 r=1 ΩP X X l

= exp(i tkintψk (η)) ds P0(η) . ∗ Z k=1 M (0,+ ) X ﬁn ∞ We now write

l ∞ (76) exp(i tk( ϕk(xi(ω)))) dP0(ω) = Z k=1 r=1 ΩP X X l ∞ = lim exp(i tk( ϕk(xi(ω)))) dPn(ω) . n →∞ Z k=1 r=1 ΩP X X 56 ALEXANDER I. BUFETOV

On the other hand, since Pn P weakly in Mﬁn(Mﬁn (0, + ) ), we have → ∞ l

(77) lim exp(i tkintψk (η)) d(s) Pn = n ∗ →∞ Z k=1 M (0,+ ) X ﬁn ∞ l = exp(i tkintψk (η)) dP . Z k=1 M (0,+ ) X ﬁn ∞ It follows that

l ∞ (78) exp(i tk( ϕk(xi(ω)))) dP0(ω) = Z k=1 r=1 ΩP X X l

= exp(i tkintψk (η)) dP. Z k=1 M (0,+ ) X ﬁn ∞ l

Since integrals of functions of the form exp(i tkintψk (η)) determine k=1 a finite Borel measure on Mfin (0, + ) uniquely,P we have s P0 = P. ∞ 0 ∗ The Third Step: The Limit Measure is Supported on ΩP . To see that 0 P0(ΩP ΩP ) = 0, set γ0(ω) = γ(ω) + (1 xk(ω)). Since the sum \ k:x (ω) 1 − k ≥ in the right-hand side is finite, the functionP γ0 is continuous on ΩP , and we have exp( γ0(ω)) dP0(ω) = lim exp( γ0(ω)) dpPn(ω). We also n ΩP − →∞ΩP − have R R

∞ ∞ exp( min(1, xi(ω))) dP0(ω) = lim exp( min(1, xi(ω))) dpPn . − n − Z i=1 →∞ Z i=1 ΩP X ΩP X 0 Since pPn(ΩP ΩP ) = 0 for all n, we have exp( γ0(ω)) dpPn(ω) = \ ΩP − ∞ R exp( min(1, xi(ω))) dpPn(ω) , for all n, it follows that exp( γ0(ω)) dP0(ω) = ΩP − i=1 ΩP − R P∞ R ∞ exp( min(1, xi(ω))) dP0(ω) , whence the equality γ0(ω) = min(1, xi(ω)) ΩP − i=1 i=1 R P ∞ P and, consequently, also the equality γ(ω) = xi(ω) holds P0-almost surely, i=1 0 and P0(ΩP ΩP ) = 0. We thus have P0 = pPP. The proof is complete. \ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 57

7. PROOFOF LEMMA 1.14 AND COMPLETIONOFTHE PROOFOF THEOREM 1.11. 7.1. Reduction of Lemma 1.14 to Lemma 7.1. Recall that we have introduced a sequence of mappings (n) 0 r : Mat(n, C) ΩP , n N → ∈ that to z Mat(n, C) assigns the point ∈ trz z λ (z) λ (z) r(n)(z) = ( ∗ , 1 ,..., n , 0,..., 0), n2 n2 n2 where λ1(z) > ... > λn(z) > 0 are the eigenvalues of the matrix z∗z, counted with multiplicities and arranged in non-increasing order. By definition, we have trz z γ(r(n)(z)) = ∗ . n2 Following Vershik [39], we now introduce on Mat(N, C) a sequence of averaging operators over the compact groups U(n) U(n). × 1 (79) (Anf)(z) = f(u1zu2− )du1du2, U(n)ZU(n) × where du stands for the normalized Haar measure on the group U(n). For any U( ) U( )-invariant probability measure on Mat(N, C), the oper- ∞ × ∞ ator An is the operator of conditional expectation with respect to the sigma- algebra of U(n) U(n)-invariant sets. × (n) By definition, the function (Anf)(z) only depends on r (z). Lemma 7.1. Let m N. There exists a positive Schwartz function ϕ on ∈ Mat(m, C) as well as a positive continuous function f on ΩP such that for any z Mat(N, C) and any n m we have ∈ ≥ (80) f(r(n)(z)) (A ϕ)(z). ≤ n Remark. The function ϕ, initially defined on Mat(m, C), is here extended to Mat(N, C) in the obvious way: the value of ϕ at a matrix z is set to be its value on its m m corner. We postpone the proof× of the Lemma to the next subsection and proceed with the the proof of Lemma 1.14. Refining the definition of the class F in the introduction, take m N and let F(m) is the family of all Borel sigma-finite U( ) U( )-invariant∈ ∞ × ∞ measures ν on Mat(N, C) such that for any R > 0 we have

ν( z : max zij < R ) < + . i,j m { ≤ | | } ∞ 58 ALEXANDER I. BUFETOV

Equivalently, the measure of a set of matrices, whose m m-corners are required to lie in a compact set, must be finite; in particular,× the projections (πn∞) ν are well-defined for all m. For example, if s+m > 0, then the Pick- ∗ rell measure µ(s) belongs to F(m). Recall furthermore that, by the results of [8], [9] any measure ν F(m) admits a unique ergodic decomposition into finite ergodic components:∈ in other words, for any such ν there exists a unique Borel sigma-finite measure ν on ΩP such that we have

(81) ν = ηωdν(ω).

ΩZP Since the orbit of the unitary group is of course a compact set, the measures (r(n)) ν are well-defined for n > m and may be thought of as finite- dimensional∗ approximations of the decomposing measure ν. Indeed, recall from the introduction that, if ν is finite, then the measure ν is the weak limit of the measures (r(n)) ν as n . The following proposition is a stronger and a more precise version∗ of→ Lemma ∞ 1.14 from the introduction.

Proposition 7.2. Let m N, let ν F(m), let ϕ and f be given by Lemma ∈ ∈ (n) 7.1, and assume ϕ L1(Mat(N, C), ν). Then f L1(ΩP , (r ) ν) for all ∈ (n) ∈ ∗ n > m, f L1(ΩP , ν) and f(r ) ν fν weakly in Mﬁn(ΩP ). ∈ ∗ → Proof. First Step: The Martingale Convergence Theorem and the Ergodic Decomposition. We start by formulating a pointwise version of the equality (27) from the Introduction: for any z Mat and any bounded continuous ∈ reg function ϕ on Mat(N; C) we have

(82) lim Anϕ(z) = fdηr (z) n ∞ →∞ Mat(ZN;C)

(here, as always, given ω ΩP , the symbol ηω stands for the ergodic probability measure corresponding∈ to ω.) Indeed, (82) immediately follows from the deﬁnition of regular matrices, the Olshanski-Vershik characterization of the convergence of orbital measures [26] and the Reverse Martingale Con- vergence Theorem. The Second Step. Now let ϕ and f be given by Lemma 7.1, and assume

ϕ L1(Mat(N, C), ν). ∈ Lemma 7.3. for any ε > 0 there exists a U( ) U( ) invariant set ∞ × ∞ − Yε Mat(N, C) such that ⊂ (1) ν(Y ) < + ; ε ∞ THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 59

(2) for all n > m we have

f(r(n)(z))dν(z) < ε.

Mat(NZ,C) Yε \ Proof. Since ϕ L1(Mat(N, C), ν), we have ∈ ( ϕdη )dν(ω) < + . ω ∞ ΩZP Mat(ZN,C) Choose a Borel subset Y˜ ε Ω in such a way that ν(Y˜ ε) < + and ⊂ P ∞

( ϕdηω)dν(ω) < ε.

Y˜Zε Mat(ZN,C) The pre-image of the set Y˜ ε under the map ω or, more precisely, the set ∞ ε Yε = z Matreg : ω (z) Y˜ { ∈ ∞ ∈ } is by deﬁnition U( ) U( ) invariant and has all the desired properties. ∞ × ∞ − The Third Step. Let ψ :ΩP R be continuous and bounded. Take ε > 0 → and the corresponding set Yε. For any z Mat we have ∈ reg lim ψ(r(n)(z)) f(r(n)(z)) = ψ(ω (z)) f(ω (z)). n ∞ ∞ →∞ · · Since ν(Y ) < , the bounded convergence theorem gives ε ∞ (83) lim ψ(r(n)(z)) f(r(n)(z))dν(z) = n →∞ · YZε = ψ(ω (z)) f(ω (z))dν(z). ∞ · ∞ YZ

By deﬁnition of Yε for all n N, n > m, we have ∈ ψ(r(n)(z)) f(r(n)(z))dν(z) ε sup ψ . | · | ≤ ΩP | | Mat(NZ,C) Yε \ It follows that

(84) lim ψ(r(n)(z)) f(r(n)(z))dν(z) = n →∞ · Mat(ZN,C) = ψ(ω (z)) f(ω (z))dν(z), ∞ · ∞ Mat(ZN,C) 60 ALEXANDER I. BUFETOV which, in turn, implies that

lim ψfd(r(n)) (ν) = ψfdν, n ∗ →∞ ΩZP ΩZP that the weak convergence is established, and that the Lemma is proved completely.

7.2. Proof of Lemma 7.1. Introduce an inner product , on Mat(m, C) h i by the formula z , z = tr(z∗z ). This inner product is naturally ex- h 1 2i < 1 2 tended to a pairing between the projective limit Mat(N, C) and the inductive limit ∞ Mat0 = Mat(m, C). m=1 [ For a matrix ζ Mat set ∈ 0 Ξζ (z) = exp(i ζ, z ), z Mat(N, C). h i ∈ We start with the following simple estimate on the behaviour of the Fourier transform of orbital measures. Lemma 7.4. Let m N. For any ε > 0 there exists δ > 0 such that for any ∈ n > m and ζ Mat(m, C), z Mat(N, C) satisfying tr(ζ∗ζ)tr((πn∞(z))∗(πn∞(z)) < δn2 we have ∈1 A Ξ (z) <∈ ε. | − n ζ | Proof. This is a simple corollary of the power series representation of the Harish-Chandra–Itzykson–Zuber orbital integral, see e.g. [13], [14], [31]. (n) (n) Indeed, let σ , , σ be the eigenvalues of z∗z, and let x , , xn be 1 ··· m 1 ··· the eigenvalues of πn∞(z). The standard power series representation, see e.g. [13], [14],[31], for the Harish-Chandra–Itzykson–Zuber orbital integral gives, for any n N, a representation ∈ x(n) x(n) A Ξ(z) = 1 + a(λ, n)s (σ , , σ ) s ( 1 , , n ), n λ 1 ··· m · λ n2 ··· n2 λ Y+ X∈ where the summation takes place over the set Y+ all non-empty Young di- agrams λ, sλ stands for the Schur polynomial corresponding to the diagram λ, and the coefficients a(λ, n) satisfy sup a(λ, n) 1. The proposition λ Y+ | | ≤ ∈ follows immediately. Corollary 7.5. For any m N, ε > 0, R > 0, there exists a positive ∈ Schwartz function ψ : Mat(m, C) (0, 1] such that for all n > m we have → (85) A ψ(π∞(z)) 1 ε n m ≥ − THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 61 for all z satisfying 2 tr((πn∞(z))∗(πn∞(z)) < Rn . Proof. Let ψ be a Schwartz function taking values in (0, 1]. Assume additionally that ψ(0) = 1 and that the Fourier transform of ψ is supported in the ball of radius ε0 around the origin. A Schwartz function satisfying all these requirements is constructed without difficulty. By Lemma 7.4, if ε0 is small enough as a function of m, ε, R, then the inequality (85) holds for all n > m.Corollary 7.5 is proved completely. We now conclude the proof of Lemma 7.1. Take a sequence R , and let ψ be the corresponding sequence n → ∞ n of Schwartz functions given by Corollary 7.5. Take positive numbers tn decaying fast enough so that the function ϕ = n∞=1 tnψn is Schwartz. Let f˜ be a positive continuous function on (0, + ) such that for any n, P ∞ if t Rn, then f˜(t) < tn/2. For ω ΩP , ω = (γ, x), set f(ω) = f˜(γ(ω)). The≤ function f is by definition positive∈ and continuous. By Corollary 7.5, the functions ϕ and f satisfy all requirements of Lemma 7.1, which, therefore, is proved completely. 7.3. Completion of the proof of Theorem 1.11.

Lemma 7.6. Let E be a locally compact complete metric space. Let Bn, B be sigma-finite measures on E, let P be a probability measure on E, and let f, g be positive bounded continuous functions on E. Assume that for all n N we have g L1(E, Bn) and that fBn fB weakly in Mfin(E), while∈ ∈ → gBn P gdBn → E weakly in Mfin(E) as n . R → ∞ g Then g L (E, ) and = B 1 B P gdB ∈ E R Proof. Let ϕ be a nonnegative bounded continuous function on E. As n , we have ϕfgdBn ϕfgdB, and, on the other hand, we have → ∞ E → E R R ϕfgdBn E (86) ϕfdP. R gdBn → E EZ Choosing ϕ = 1, we obtainR that fgdB E lim gdBn = > 0; n R →∞ fdP EZ E R 62 ALEXANDER I. BUFETOV the sequence gdBn is thus bounded away both from zero and infinity. E Furthermore, forR arbitrary bounded continuous positive ϕ we have ϕfgdB E (87) lim gdBn = . n R →∞ ϕfdP EZ E R Now take R > 0 and ϕ(x) = min(1/f(x),R). Letting R tend to , ∞ we obtain lim gdBn = gdB. Substituting into (86), we arrive at the n →∞ E E equality R R ϕfgdB E ϕfdP = . R gdB EZ E Note that here, as in (86), the function ϕR may be an arbitrary nonnegative continuous function on E. In particular, taking a compactly supported func- ψgdB tion ψ on E and setting ϕ = ψ/f, we obtain ψd = E . Since this P R gdB E E equality is true for any compactly supported fnctionR ψ onRE, we conclude that = gB , and the Lemma is proved completely. P gdB E CombiningR Lemma 7.6 with Lemma 1.14 and Proposition 1.16, we conclude the proof of Theorem 1.11. Theorem 1.11 is proved completely.

7.4. Proof of Proposition 1.4. In view of Proposition 1.10 and Theorem 1.11, it sufﬁces to prove the singularity of the ergodic decomposition measures µ(s1), µ(s2). Since, by Proposition 1.9, the measures µ(s1), µ(s2) are mutually singular, there exists a set D Mat(N, C) such that ⊂ µ(s1)(D) = 0, µ(s2)(Mat(N, C) D) = 0. \ ˜ Introduce the set D = z Mat(N, C) : lim AnχD(z) = 1 . By def- n { ∈ →∞ } inition, the set D˜ is U( ) U( )-invariant, and we have µ(s1)(D˜) = (s2) ˜ ∞ × ∞ 0, µ (Mat(N, C) D) = 0. Introduce now the set D ΩP by the \ ⊂ formula D = ω Ω : η (D˜) = 1 . We clearly have µ(s1)(D) = { ∈ P ω } 0, µ(s2)(Ω D) = 0. Proposition 1.4 is proved completely. P \

8. CONSTRUCTIONOF PICKRELL MEASURESANDPROOFOF PROPOSITIONS 1.8 AND 1.9. 8.1. Proof of Proposition 1.8. First we recall that the Pickrell measures are naturally deﬁned on the space of rectangular m n-matrices. Let × THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 63

Mat(m n, C) be the space of m n matrices with complex entries: × × Mat(m n, C) = z = (zij), i = 1, . . . , m; j = 1, . . . , n × { } Denote dz the Lebesgue measure on Mat(m n, C). Take s R. Let × ∈ m0, n0 be such that m0 + s > 0, n0 + s > 0. Following Pickrell, take (s) m > m0, n > n0 and introduce a measure µm,n on Mat(m n, C) by the formula × (s) (s) m n s (88) µ = const det(1 + z∗z)− − − dz, m,n m,n · × where m (s) mn Γ(l + s) (89) const = π− . m,n · Γ(n + l + s) l=Ym0 m,n For m1 m, n1 n, let πm1,n1 : Mat(m n, C) Mat(m1 n1, C) be the natural≤ projection≤ map. × → × Proposition 8.1. Let m, n N be such that s > m 1. Then for any ∈ − − z˜ Mat(n, C) we have ∈ m n 1 s (90) det(1 + z∗z)− − − − dz =

m+1,nZ 1 (πm,n )− (˜z)

n Γ(m + 1 + s) m n s π det(1 +z ˜∗z˜)− − − . Γ(n + m + 1 + s) Proposition 1.8 is an immediate corollary of Proposition 8.1. Proof of Proposition 8.1. As we noted in the Introduction, the following computation goes back to the classical work of Hua Loo Keng [18]. Take z Mat (m+ ∈ 1) n, C . Multiplying, if necessary, by a unitary matrix on the left and on × the right, represent the matrix πm+1,n z = z in diagonal form, with positive m, n real entries on the diagonal: z = u > 0, i = 1, . . . , n, z = 0 for i = j. ii i ij 6 Here we set ui = 0 for i > min(n, m). Denotee ξi = zm+1,i, i = 1, . . . , n. Write e e m n 2 2 m 1 n s 2 m 1 n s ξi ui m 1 n s det(1+z∗z)− − − − = (1+u )− − − − (1+ξ∗ξ | | )− − − − . i × − 1 + u2 i=1 i=1 i Y X We have n 2 2 n 2 ξi ui ξi 1 + ξ∗ ξ | | = 1 + | | . − 1 + u2 1 + u2 i=1 i i=1 i X X Integrating in ξ, we ﬁnd + n 2 m n ∞ ξi m 1 n s 2 π n 1 m 1 n s (1+ | | )− − − dξ− = (1+u ) r − (1+r)− − − − dr , 1 + u2 i Γ(n) i=1 i i=1 Z X Y Z0 64 ALEXANDER I. BUFETOV where n ξ 2 (91) r = r(m+1,n)(z) = | i| . 1 + u2 i=1 i X Recalling the Euler integral + ∞ n 1 m 1 n s Γ(n) Γ(m + 1 + s) (92) r − (1 + r)− − − − dr = · , Γ(n + 1 + m + s) Z0 we arrive at the desired conclusion. Furthermore, introduce a map m+1,n π m, n : Mat (m + 1) n, C Mat m n, C R+ × −→ × × by the formula e m+1,n m+1,n (m+1,n) π m, n (z) = (π m, n (z), r (z)) , where r(m+1,n)(z) is given by the formula (91). (m,n,s) e Let P be a probability measure on R+ given by the formula:

(m,n,s) Γ(n + m + s) n 1 m n s dP (r) = r − (1 r)− − − dr . Γ(n) Γ(m + s) − · The measure P (m,n,s) is well-deﬁned as soon as m + s > 0. Corollary 8.2. For any m, n N and s > m 1, we have ∈ − − m+1,n (s) (s) (m+1,n,s) (π m, n ) µm+1,n = µm,n P . ∗ × Indeed, this is precisely what was shown by our computation. Removing a columne is similar to removing a row:

m,n+1 t m+1,n t (π m, n (z)) = π m, n (z ). Write r(m,n+1)(z) = r(n+1,m)(zt). Introduce a map m,n+1 π m, n : Mat m (n + 1), C Mat m n, C R+ e × −→ × × by the formula e m,n+1 m,n+1 (m,n+1) π m, n (z) = (π m, n (z), r (z)) . Corollary 8.3. For any m, n N and s > m 1, we have ∈ − − e m,n+1 (s) (s) e (n+1,m,s) (π m, n ) µm,n+1 = µm,n P . ∗ × Now take n such that n + s > 0 and introduce a map e πn : Mat N N, C Mat n n, C × −→ × by the formula , e (n+1,n) (n+1,n+1) (n+2,n+1) (n+2,n+2) πn(z) = (π∞n,∞ n (z), r , r , r , r ,...) .

e e e THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 65

We can now reformulate the result of our computations as follows: Proposition 8.4. If n + s > 0, then we have

∞ (s) (s) (n+l+1,n+l,s) (n+l+1,n+l+1,s) (93) (πn) µ = µm,n (P P ) . ∗ × × Yl=0 8.2. Proofe of Proposition 1.9. Using Kakutani’s theorem, we now conclude the proof of Proposition 1.9. Take n large enough so that n + s > 1, n + s0 > 1 and compute the Hellinger integral

(n,n 1,s) (n,n,s) (n,n 1,s ) (n,n,s ) Hel (n, s, s0) = E( (P − P ) (P − 0 P 0 )) = × · × q Γ(2n 1 + s) Γ(2n 1 + s ) Γ(2n + s) Γ(2n + s ) = − − 0 0 Γ(n 1)Γ(n + s) · Γ(n 1)Γ(n + s ) · Γ(n)Γ(n + s) · Γ(n)Γ(n + s )× s − − 0 0 ∞ ∞ n 1 2n 1 s+s0 n 1 2n s+s0 r − (1 + r)− − − 2 dr r − (1 + r)− − 2 dr = × · Z0 Z0

s+s0 2 Γ(2n 1 + s) Γ(2n 1 + s0) Γ(2n + s) Γ(2n + s0) (Γ(n + 2 )) = − · s+s− · s+s . Γ(2n 1 + 0 ) · Γ(2n + 0 ) ·Γ(n + s) Γ(n + s0) p − 2 p 2 · We now recall a classical asymptotics: as t , we have → ∞ Γ(t + a ) Γ(t + a ) (a + a )2 1 1 · 2 = 1 + 1 2 + O( ). a1+a2 2 2 (Γ(t + 2 )) 4t t It follows that 2 (s + s0) 1 Hel (n, s, s0) = 1 + O( ) , − 8 n n2 whence, by the Kakutani’s theorem combined with (93), the Pickrell mea- (s) (s ) sures µ and µ 0 , finite or infinite, are mutually singular if s = s0. 6 REFERENCES [1] E.M. Baruch, The classical Hankel transform in the Kirillov model, arxiv:1010.5184v1. [2] V.I. Bogachev, Measure theory. Vol. II. Springer Verlag, Berlin, 2007. [3] A.M. Borodin, Determinantal point processes, in The Oxford Handbook of Random Matrix Theory, Oxford University Press, 2011. [4] A. Borodin, A. Okounkov; G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481–515. [5] A. Borodin, G. Olshanski, Infinite random matrices and ergodic measures. Comm. Math. Phys. 223 (2001), no. 1, 87–123. [6] A.M. Borodin, E.M. Rains, Eynard-Mehta theorem, Schur process, and their pfaffian analogs. J. Stat. Phys. 121 (2005), 291–317. 66 ALEXANDER I. BUFETOV

[7] P. Bourgade, A. Nikeghbali, A. Rouault, Ewens Measures on Compact Groups and Hypergeometric Kernels, Seminaire´ de Probabilites´ XLIII, Springer Lecture Notes in Mathematics 2011, pp 351-377. [8] A.I. Bufetov, Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups, arXiv:1105.0664, May 2011, to appear in Matematich- eskii Sbornik . [9] A.I. Bufetov, Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices, arXiv:1108.2737, August 2011, to appear in Annales de l’Institut Fourier. [10] A.I. Bufetov, Multiplicative functionals of determinantal processes, Uspekhi Mat. Nauk 67 (2012), no. 1 (403), 177–178; translation in Russian Math. Surveys 67 (2012), no. 1, 181–182. [11] A. Bufetov, Infinite determinantal measures, Electronic Research Announcements in the Mathematical Sciences, 20 (2013), pp. 8 – 20. [12] D.J.Daley, D. Vere-Jones, An introduction to the theory of point processes, vol.I-II, Springer Verlag 2008. [13] J. Faraut, Analyse sur les groupes de Lie: une introduction, Calvage et Mounet 2006. [14] J. Faraut, Analysis on Lie Groups, Cambridge University Press, 2008. [15] S. Ghobber, Ph. Jaming, Strong annihilating pairs for the Fourier-Bessel transform. J. Math. Anal. Appl. 377 (2011), 501–515. [16] S. Ghobber, Ph. Jaming, Uncertainty principles for integral operators, arxiv:1206.1195v1 [17] J.B. Hough, M. Krishnapur, Y. Peres, B. Virag,´ Determinantal processes and inde- pendence. Probab. Surv. 3 (2006), 206–229. [18] Hua Loo Keng, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Science Press Peking 1958, Russian translation Moscow Izd.Inostr. lit., 1959, English translation (from the Russian) AMS 1963. [19] A. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer Verlag, 1933. [20] A. Lenard, States of classical statistical mechanical systems of infinitely many particles. I. Arch. Rational Mech. Anal. 59 (1975), no. 3, 219–239. [21] R. Lyons, Determinantal probability measures. Publ. Math. Inst. Hautes Etudes´ Sci. No. 98 (2003), 167–212. [22] E. Lytvynov, Fermion and boson random point processes as particle distributions of infinite free Fermi and Bose gases of finite density. Rev. Math. Phys. 14 (2002), no. 10, 1073–1098. [23] O. Macchi, The coincidence approach to stochastic point processes. Advances in Appl. Probability, 7 (1975), 83–122. [24] Yu.A. Neretin, Hua-type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114 (2002), no. 2, 239–266. [25] G. Olshanski, The quasi-invariance property for the Gamma kernel determinantal measure. Adv. Math. 226 (2011), no. 3, 2305–2350. [26] G. Olshanski, A. Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. In: Contemporary mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, 1996, pp. 137–175. [27] D. Pickrell, Mackey analysis of infinite classical motion groups. Pacific J. Math. 150 (1991), no. 1, 139–166. [28] D. Pickrell, Separable representations of automorphism groups of infinite symmetric spaces, J. Funct. Anal. 90 (1990), 1–26. THE ERGODIC DECOMPOSITION OF INFINITE PICKRELL MEASURES 67

[29] D. Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70 (1987), 323–356. [30] M. Rabaoui, Asymptotic harmonic analysis on the space of square complex matrices. J. Lie Theory 18 (2008), no. 3, 645–670. [31] M. Rabaoui, A Bochner type theorem for inductive limits of Gelfand pairs. Ann. Inst. Fourier (Grenoble), 58 (2008), no. 5, 1551–1573. [32] M. Reed, B. Simon, Methods of modern mathematical physics. vol.I-IV. Second edi- tion. Academic Press, Inc. New York, 1980. [33] T. Shirai, Y. Takahashi, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes. J. Funct. Anal. 205 (2003), no. 2, 414–463. [34] T. Shirai, Y. Takahashi, Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their ergodic and Gibbs properties. Ann. Probab. 31 (2003), no. 3, 1533–1564. [35] B. Simon, Trace class ideals, AMS, 2011. [36] A. Soshnikov, Determinantal random point fields. (Russian) Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160; translation in Russian Math. Surveys 55 (2000), no. 5, 923–975. [37] G. Szego,¨ Orthogonal polynomials, AMS 1969. [38] C. A. Tracy, H. Widom, Level spacing distributions and the Bessel kernel. Comm. Math. Phys. 161, no. 2 (1994), 289–309. [39] A.M. Vershik, A description of invariant measures for actions of certain infinite- dimensional groups. (Russian) Dokl. Akad. Nauk SSSR 218 (1974), 749–752.

AIX-MARSEILLE UNIVERSITE´,CNRS,CENTRALE MARSEILLE, I2M, UMR 7373

STEKLOV INSTITUTEOF MATHEMATICS,MOSCOW

INSTITUTEFOR INFORMATION TRANSMISSION PROBLEMS,MOSCOW

NATIONAL RESEARCH UNIVERSITY HIGHER SCHOOLOF ECONOMICS,MOSCOW

RICE UNIVERSITY,HOUSTON TX