Probability, Geometry and Integrable Systems MSRI Publications Volume 55, 2008
Nonintersecting Brownian motions, integrable systems and orthogonal polynomials
PIERRE VAN MOERBEKE
To Henry, teacher and friend, with admiration and gratitude
ABSTRACT. Consider n nonintersecting Brownian motions on ޒ, which leave from p definite points and are forced to end up at q points at time t 1. When n , the equilibrium measure for these Brownian particles hasD its support on!p 1intervals, for t 0, and on q intervals, for t 1. Hence it is clear that, when t evolves, intervals must merge, must disappear and be created, leading to various phase transitions between times t 0 and 1. Near these moments of phase transitions,D there appears an infinite-dimen- sional diffusion, a Markov cloud, in the limit n , which one expects to depend only on the nature of the singularity associated% 1 with this phase change. The transition probabilities for these Markov clouds satisfy nonlinear PDE’s, which are obtained from taking limits of the Brownian motion model with finite particles; the finite model is closely related to Hermitian matrix inte- grals, which themselves satisfy nonlinear PDE’s. The latter are obtained from investigating the connection between the Karlin-McGregor formula, moment matrices, the theory of orthogonal polynomials and the associated integrable systems. Various special cases are provided to illustrate these general ideas. This is based on work by Adler and van Moerbeke.
Mathematics Subject Classification: Primary: 60G60, 60G65, 35Q53; secondary: 60G10, 35Q58. Keywords: Dyson’s Brownian motion, Pearcey process, random matrices coupled in a chain, random matrices with external potential, infinite-dimensional diffusions. This work was done while PvM was a Miller visiting Professor at the University of California, Berkeley. The support of National Science Foundation grant #DMS-04-06287, a European Science Foundation grant (MISGAM), a Marie Curie Grant (ENIGMA), FNRS, Francqui Foundation and IUAP grants is gratefully acknowledged.
373 374 PIERREVANMOERBEKE
1. Introduction
This lecture in honor of Henry McKean forms a step in the direction of un- derstanding the behavior of nonintersecting Brownian motions on ޒ (Dyson’s Brownian motions), when the number of particles tends to . It explains a 1 novel interface between diffusion theory, integrable systems and the theory of orthogonal polynomials. These subjects have been at the center of Henry McK- ean’s oeuvre. I am delighted to dedicate this paper to Henry, teacher and friend, with admiration for his pioneering work in these fields. Consider n Brownian particles leaving from points a1 < < ap and forced to end up at b1 < < bq at time t 1. It is clear that, when n , the D ! 1 equilibrium measure for t 0 has its support on p intervals and for t 1 on q intervals. It is also clear that, when t evolves, intervals must merge, must disappear and be created, leading to various phase transitions, depending on the respective fraction of particles leaving from the points ai and arriving at the points bj . Therefore the region R in the space-time strip .x; t/ formed by the support . ޒ/ of the equilibrium measure as a function of time 0 t 1 will typically present singularities of different types. Near the moments, where a phase transition takes place, one would expect to find in the limit n an infinite-dimensional diffusion, a Markov cloud, % 1 having some universality properties. Universality here means that the infinite- dimensional diffusion is to depend on the type of singularity only. These Markov clouds are infinite-dimensional diffusions, which ‘in principle’ could be de- scribed by an infinite-dimensional Laplacian with a drift term. We conjecture that each of the Markov clouds obtained in this fashion is related to some in- tegrable system, which enables one to derive a nonlinear (finite-dimensional) PDE, satisfied by the joint probabilities. The purpose of this lecture is to show the intimate relationship between these subjects: nonintersecting Brownian mo- tions and integrable systems, via the theory of orthogonal polynomials. Special cases have also shown an intimate connection between the integrable system and the Riemann-Hilbert problem associated with the singularity. These ideas will then be applied to a simple model, where we show that the transition probabilities for the infinite-dimensional Brownian motions near a cusp satisfy a nonlinear PDE. The interrelations between all such equations, “initial” and “final” (t ) conditions, are interesting and challenging open problems. ! ˙1 Universality in this context is a largely open field. For references, see later. BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 375
2. Biorthogonal polynomials and the 2-component KP hierarchy
Consider the inner product for the weight .x; y/ on ޒ2,
f g f.x/g.y/.x; y/dxdy: h j i WD ޒ2 ZZ and an inner product for this weight, augmented with an extra-exponential factor, depending on “time” parameters t .t1; t2;:::/ and s .s1; s2;:::/, WD WD 1 i i 1 .ti y si x / f g t;s f.x/g.y/.x; y/e dxdy: h j i WD ޒ2 ZZ P .1/ .2/ Construct monic biorthogonal polynomials pm .y/ and pn .x/ (also depending on the parameters t and s) with regard to this deformed weight,
1 i 1 i .2/ 1 si x .1/ 1 ti y pn .x/e pm .y/e P P D ˇ E 1 i i ˇ .2/ .1/ 1 .ti y si x / pn .x/pm .y/.x; y/e dx dy ˇ D ޒ2 ZZ P ınmhn; D and let n be the determinant of the moment matrix
1 i 1 i k 1 si x ` 1 ti y n.t; s/ det x e y e : WD P P 0 k;` n 1 D ˇ E ˇ The following theorem and its corollary,ˇ due to Adler and van Moerbeke [1997; 1999b] and inspired by Sato’s theory, establishes a link between the functions n and the biorthogonal polynomials:
THEOREM 2.1. Given these data, the determinant n.t; s/ and the biorthogonal polynomials are related by the following relations, where we have set Œ˛ WD .˛; 1 ˛2; 1 ˛3;:::/ for ˛ ރ: 2 3 2 1 n n.t Œz ; s/ .1/ z pn .z/; n.t; s/ D 1 n n.t; s Œz / .2/ z C pn .z/; n.t; s/ D 1 .2/ .t Œz ; s/ p .x/ 1 i i n 1 n 1 n 1 .ti y si x / z C C e .x; y/ dx dy; n.t; s/ D ޒ2 z y ZZ P 1 .1/ .t; s Œz / p .y/ 1 i i n 1 n 1 n 1 .ti y si x / z C e .x; y/ dx dy; (2-1) n.t; s/ D ޒ2 z x ZZ P 376 PIERREVANMOERBEKE with the n.t; s/ satisfying bilinear equations, for all integers n; m 0 and all t; t ; s; s ރ : 0 0 2 1 1 0 i 1 1 1 .ti t /z n m 2 n 1.t Œz ; s/ m 1.t 0 Œz ; s0/e i z dz z C C P I D1 1 0 i 1 1 1 .si s /z m n n.t; s Œz / m.t 0; s0 Œz /e i z dz: D z C P I D1 Two-component KP hierarchy. Define the Hirota symbol between functions f f.t1; t2;:::/ and g g.t1; t2;:::/ by D D @ @ @ @ p ; ;::: f g p ; ;::: f.t y/g.t y/ : @t1 @t2 ı WD @y1 @y2 C y 0 ˇ D ˇ 1 t zi ˇ i The elementary Schur polynomials S are defined by e 1 i Si.t/z ` WD i 0 for ` 0 and S .t/ 0 for ` < 0; moreover, set for laterP use ` D P @ 1 @ 1 @ S`.@t / S` ; ; ; ::: : Q WD @t1 2 @t2 3 @t3
Finally, recall that the Wronskian f; g x of f and g is given by f g @f @g g.x/ f.x/: @x @x
COROLLARY. From Theorem 2.1, one deduces the equations
2 S @ 1 @ 2 @ j ; ;::: n 1 n 1 n log n; @t1 2 @t2 C ı D @s1@tj 1 C (2-2) 2 S @ 1 @ 2 @ j ; ;::: n 1 n 1 n log n; @s1 2 @s2 ı C D @t1@sj 1 C and finally a single partial differential equation for n in terms of Wronskians,
@2 log @2 log @2 log @2 log n ; n n ; n 0: (2-3) @t1@s2 @t1@s1 C @s1@t2 @t1@s1 D ( )t1 ( )s1
SKETCH OF PROOF OF THEOREM 2.1 AND ITS COROLLARY. The following double integral can be expanded in two different ways with regard to the param- eters a .a1; a2;:::/: WD BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 377
n.t;s/ n 1.t 0;s0/ C .2/ 1 i 0 i .1/ 1 .ti y si x / dx dy pn 1.t 0; s0 x/pn .t; s y/e .x;y/ ޒ2 I I t t a ZZ C P ˇ t‘0 t a ˇ 0‘ C ˇ s s ˇ D 1 @ 1 @ 1 @ 2 2aj 1Sj ; ; ;::: n 2 n O.a / D C @t1 2 @t2 3 @t3 C ı C j 0 XD 2 1 2 @ 2 2aj 1 n 1 log n 1 O.a / ; (2-4) D C C @s1@tj 1 C C j 0 C XD using the fact that the space H span zi; i ޚ can be equipped with two WD f 2 g (formal) inner products:
(i) f; g f.z/g.z/ dz; h iD ޒ Z i H (ii) a residue pairing about z between f i 0 aiz C and h j 1 H D 1 D 2 D j ޚ bj z : 2 2 P P dz f; h f.z/h.z/ aibi: h i1 D z 2i D I D1 i 0 X The two inner products are related by g.u/ f; g f.z/g.z/ dz f; du : h iD ޒ D ޒ z u Z Z 1 Then the two expansions (2-4) are obtained, using the -function representa- tion (2-1) of the biorthogonal polynomials, transforming the double integral (2-4) into a contour integral about and finally computing the residues. Upon 1 equating the two series in (2-4) for arbitrary aj , one finds the first identity (2-2). Application of a similar shift s s a, s s a, t t yields the second ‘ 0 ‘ C 0 D identity (2-2). Then combining the identities (2-2) for j 0 and 1 leads to the D PDE (2-3). ˜
3. Orthogonal polynomials with regard to several weights and the n-component KP hierarchy
Now considering two sets of weights,
1;:::; q and '1;:::;'p; and deform each weight with its own set of times:
1 i 1 i s 1 ski x t 1 tki y k .x/ k .x/e and 'k .y/ 'k .y/e ; WD P WD P 378 PIERREVANMOERBEKE the time parameters being
s .s ; s ;:::/ for 1 k q and t .t ; t ;:::/ for 1 k p: k D k1 k2 k D k1 k2
Take a moment matrix consisting of p q blocks of sizes mi nj , formed of moments with regard to all the different combinations of i and 'j ’s; of course for the full matrix to be a square matrix, the integers m1; m2;::: 0 and q p n1; n2;::: 0 must satisfy mi ni. Define the determinant mn of 1 D 1 these moment matrices (the inner product is the same as in Section 2): P P
m1;:::;mq n1;:::;np .s1;:::; sq t1;:::; tp/ I I WD k s1 ` t1 k s1 ` tp x 1 .x/ y '1 .y/ 0k .t z 1 / n` mn ` .``/ n` z Qmn .z/ z ; mn WD D C .t z 1 / n˛ 1 m;n e` e˛ ˛ .`˛/ n˛ 1 z C Qmn .z/ c˛z for ˛ ` mn D D C ¤ p q are polynomials (involving 1 n˛ coefficients), satisfying 1 m˛ orthogonal- ity conditions P P p j s .`i/ t 1 ˛ q; x ˛ .x/ Qmn .y/'i .y/ 0 for D 0 j m˛ 1: ˇ i 1 ( ˇ XD ˇ II. Similarly, the expressionsˇ 1 m˛ 1 m e˛ ;n e` .s˛ Œz / .`˛/ z C Pnm .z/ of degree < m˛ ˙ mn D BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 379 q p are polynomials (involving 1 m˛ coefficients), satisfying 1 n˛ orthogonal- ity conditions: P P q 1 ˛ p; 0 j n˛ 1 P .`i/.x/ s.x/ yj 't .y/ 0 for nm i ˛ D except ˛ `; j n 1; i 1 ˇ D D ` XD ˇ q ˇ P .`i/.x/ s.x/ ˇ yn` 1't .y/ 1: nm i ` D i 1 ˇ XD ˇ ˇ III. The Cauchy transformsˇ of the polynomials in II are q .t Œz 1/ 't .y/ n` mn ` .`i/ s ` z C Pnm .x/ i .x/ ; mn WD z y i 1 ˇ XD ˇ q ˇ .t Œz 1/ ˇ 't .y/ n` 1 m;n e` e˛ ` .˛i/ s ` z C C Pnm .x/ i .x/ : ˙ mn D z y i 1 ˇ XD ˇ ˇ IV. The Cauchy transforms of the polynomials in I are ˇ 1 s p m˛ 1 m e˛;n e` .s˛ Œz / ˛ .x/ .`i/ t z C C Qmn .y/'i .y/ : ˙ mn D z x ˇ i 1 ˇ XD ˇ The orthogonality conditions for these polynomials leadˇ to the following state- ment: PROPOSITION 3.1. The determinants mn defined in (3-1) satisfy the .p q/-KP C hierarchy; that is, p 1 1 1 .t t 0 /zi n n0 2 0 0 1 ˇi ˇi ˇ ˇ m;n eˇ .tˇ Œz / m ;n eˇ .tˇ0 Œz /e z dz C C P D ˇ 1 I1 XD q 1 1 1.s s0 /zi m0 m 2 0 0 1 ˛i ˛i ˛ ˛ m e˛;n.s˛ Œz / m e˛;n .s˛0 Œz /e z dz; ˙ C C P ˛ 1 I1 XD where m n 1 and m˛ n˛ 1. ˛0 D ˛0 C D C These polynomialsP P happen to beP the so-calledP multiple orthogonal polynomials of mixed type, introduced in [Daems and Kuijlaars 2007] in the context of nonin- tersecting Brownian motions; they generalize multiple orthogonal polynomials, introduced in [Aptekarev 1998; Aptekarev et al. 2003; Adler and van Moerbeke 1999a]. This will now be used in the next section. 380 PIERREVANMOERBEKE 4. Nonintersecting Brownian motions If the transition density for standard Brownian motion x.t/ in ޒ, leaving from x and arriving at y, is given by 1 .x y/2=t p.t; x; y/ e ; D pt then the probability that N nonintersecting Brownian motions x1.t/;:::; xN .t/ in ޒ, leaving at ˛ .˛1;:::;˛N / and arriving at ˇ .ˇ1;:::;ˇN /, belong WD WD to E at time t, is given by the Karlin–McGregor formula [1959]: N det p.t;˛i; xj / 1 i;j N det p.1 t; xi; ˇj / 1 i;j N dxi: EN Z i 1 YD Considering the particular case where several points coincide, i.e., where m1 m2 mq N ˛ a .a1; a1;:::; a1; a2; a2;:::; a2;:::; aq; aq;:::; aq/ ޒ WD D 2 (4-1) ‚ …„n1 ƒ ‚ …„n2 ƒ ‚ …„np ƒ N ˇ b .b1; b1;:::; b1; b2; b2;:::; b2;:::; bp; bp;:::; bp/ ޒ ; WD D 2 one verifies that‚ the…„ probabilityƒ ‚ below…„ canƒ be expressed‚ …„ as aƒ determinant of a moment matrix of the form (3-1) with p q blocks, .x1.0/;:::; xN .0// ˛ ސ all xi.t/ E D .0 < t < 1/ 2 .x .1/;:::; x .1// ˇ ˇ 1 N ˇ D lim ˇ D .˛1;:::;˛N / aˇ .ˇ1;:::;ˇN /!b ! N 1 detŒp.t;˛i; xj /1 i;j N det p.1 t; xi ; ˇj / 1 i;j N dxi ZN EN Z i 1 YD N ! y2 i j .a˛ bˇ/y det dy e 2 y C e Q C Q ; (4-2) Z 0 i 2 2.1 t/ 2t E E ; ai ai; bi bi: Q D t.1 t/ Q D t Q D 1 t s r r PROOF. It is based on the matrix identity ˜ det .Aik /1 i;k n det .Bik /1 i;k n det Ai;.j/ Bj;.j/ 1 i;j n : D Sn X2 BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 381 Upon adding extra-time parameters t .t ; t ;:::/ and s˛ .s˛;1; s˛;2;:::/ ˇ D ˇ;1 ˇ;2 D to y2 i j .a˛ bˇ/y det dy e 2 y C e Q C Q ; 0 i m1;:::;mq n1;:::;np .t1;:::; tp s1;:::; sq/ I I 2 y 1 k i j .a˛ bˇ/y 1 .tˇ;k s˛;k /y det dy e 2 y C e Q C Q C D 0 i Consider n n1 n2 nonintersecting Brownian motions on ޒ, all leaving D C from the origin, with n1 paths forced to go to a and n2 paths forced to go to a, at time t 1. The probability that all the particles belong to the set E at D time 0 < t < 1 can be expressed as a Gaussian Hermitian random matrix “with external potential”, specified by the diagonal matrix ˛ :: 0 : O 1 n1 ˛ l 2t A B C with ˛ a ; WD B ˛ C D 1 t B C n2 r B : C l B O :: C B C B ˛ C B C @ A but also as a determinant of a moment matrix, a consequence of Section 4. This 382 PIERREVANMOERBEKE gives (with n n1 n2), D C all xj .0/ 0, ސ a D 0˙ all xi.t/ E n1 left paths end up at a at time t 1, 0 2 ˇ D 1 ˇ n2 right paths end up at a at time t 1 ˇ C D @ ˇ A ˇ 2t 2 ސn a E ; (5-1) D 1 t I t.1 t/ r r with ސn being an integral over the space Hn.E0/ of Hermitian matrices with spectrum belonging to the set E ޒ: 0 1 Tr. 1 M 2 AM / ސn.˛ E0/ dM e 2 I WD Z H 0 n Z n.E / i j 1 z2=2 ˛z z C e C dz 0 1 i n1; E 1 0Z 1 j n1 n2 1 det C (5-2) D Zn B C B i j 1 z2=2 ˛z C B z C e dz C B E0 1 i n2; C BZ 1 j n1 n2 C B C C @ A THEOREM 5.1 [Adler and van Moerbeke 2007]. The log of the probability ސn.˛ E/ satisfies a fourth-order PDE in ˛ and in the boundary points b1;:::; I b2r of the set E, with quartic nonlinearity: F C F 0 det B 1F C B 1F F GC F CG 0; (5-3) 0 C 1 D B2 B2 B B 1F C 1F F 1GC F C 1G @ C A B 2r k 1 where k i 1 bi C @=@bi and WD D P @ F C 2B 1 B 1 log ސn 4n1; F F C ˛ ˛ WD @˛ D n1! n2 $ ˇ 2GC H1C; F C B H2C; F C @=@˛ ; G GCˇ ˛ ˛ ; WD 1 D n1! n2 $ ˚ « ˚ « ˇ with ˇ @ @ @ n B B ސ B B ސ 2 H1C 0 ˛ ˛ 1 log n 0 1 4 log n 4n1 ˛ ; WD @˛ @˛ C C @˛ C C ˛ @ @ B B ސ B B B ސ H2C 0 ˛ ˛ 1 log n . 0 2˛ 1 2/ 1 log n: WD @˛ @˛ SKETCH OF PROOF. In view of the results in Section 3, we add extra parameters t1; t2;::: , s1; s2;::: and ˇ to the integrals in the moment matrix above (5-2). In terms of the Vandermonde determinants k .x/ 1 i variables x1;:::; xn1 and n.x; y/ for all variables x1;:::; xn1 ; y1;:::; yn2 , we obtain from the results in Section 4 that (again with n n1 n2) the function D C .ijC.t; s;˛;ˇ; E//1 i n1; 1 j n1 n2 n1n2 .t;s;u ˛;ˇ E/ det C I I WD .ij .t; u;˛;ˇ; E//1 i n2; 1 j n1 n2 ! C n1 n2 1 1 i 1 i 1 ti x 1 ti y n.x; y/ e j e j D n1! n2! En Z j 1 P j 1 P YD YD n1 2 2 1 i x =2 ˛xj ˇx 1 si x n .x/ e j C C j e j dxj 1 j 1 P YD n2 2 2 1 i y =2 ˛yj ˇy 1 ui y n .y/ e j j e j dyj (5-4) 2 j 1 P YD satisfies the 3-component KP equation, since p q 2 1 3, since this matrix C D C D corresponds to p 2; q 1. The function n n .t; s; u ˛; ˇ E/ also satisfies D D 1 2 I I Virasoro constraints, to be explained below. (i) The three-component KP bilinear equations of Proposition 3.1 imply, using a standard residue computation on the bilinear equation (equations of the type (2-2) for j 0 and j 1, except that the three-component KP bilinear D D equations give rise to -functions depending on two integer indices) 2 @ log n1;n2 n1 1;n2 n1 1;n2 C (5-5) @t @s 2 1 1 D n1;n2 and 2 @ n1 1;n2 .@ =@t2@s1/ log n1;n2 log C (5-6) 2 @t1 n1 1;n2 D.@ =@t1@s1/ log n1;n2 2 @ n1 1;n2 .@ =@t1@s2/ log n1;n2 log C : (5-7) 2 @s1 n1 1;n2 D.@ =@t1@s1/ log n1;n2 (ii) The Virasoro equations are as follows: The integral n n .t;s;u ˛;ˇ E/, 1 2 I I as defined in (5-4), satisfies n1;n2 Bm n ;n ޖ n ;n for m 1; (5-8) 1 2 D m 1 2 where Bm and ޖm are differential operators: 2r 2r @ B m 1 ޒ m bi C ; for E Œb2i 1; b2i D @bi D X1 [1 384 PIERREVANMOERBEKE and (with the convention that ti is omitted whenever it appears for i 0; 1;::: ) D 1 @2 @2 @2 n1n2 ޖm WD 2 @ti@tj C @si@sj C @ui@uj i j m CXD @ @ @ iti isi iui C @ti m C @si m C @ui m i 1 C C C X @ @ .n1 n2/ . m/t m n1 . m/s m C C @tm C @sm C @ 2 2 n2 . m/u m .n1 n1n2 n2/ım0 @um C C C C m.m 1/ ˛.n1 n2/ım 1;0 C .t m s m u m/ C C C 2 C C @ @ @ ˛ .m 1/.s m 1 u m 1/ @tm 2 C @sm 1 C @um 1 C C C C C @ @ C C 2ˇ : C @um 2 @sm 2 C C These Virasoro equations are obtained by setting m 1 xi xi "x ; ‘ C i C m 1 yi yi "y ‘ C i C in the integral (5-4) and observing that this substitution does not change the value of the integral, provided the boundary is changed infinitesimally as well. The Virasoro constraints (5-8) above for m 1 and m 0 lead to the fol- D D lowing equations for f log n n .t; s; u ˛; ˇ E/ along the locus L of points D 1 2 I I where t s u 0, ˇ 0: D D D D @f B 1f ˛.n1 n2/; @t1 D C @f 1 @ ˛ B 1 f .n2 n1/; @s D 2 @˛ C 2 1 @2f @ 2 B 1 B 1 f 2n1; @t @s D @˛ 1 1 @2f @ @ @f 2 ˛ B0 1 B 1f 2 2˛.n1 n2/; @t @s D @˛ C @ˇ C @˛ 1 2 @2f @ @ 2 .B0 ˛ ˛B 1/f B 1.B0 1/f 2˛.n1 n2/: (5-9) @t2@s1 D @˛ @˛ C BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 385 From the differential equations (5-6)–(5-7) and from the two first two Virasoro equations (5-9) it follows that, along the locus L, and for the indices n1 1; n2, ˙ @2 log n1n2 @t2@s1 @ n1 1;n2 n1 1;n2 log C B log C 2˛; 2 1 @ D @t1 n1 1;n2 D n1 1;n2 C log n1n2 @t1@s1 @2 log n1n2 @t1@s2 @ n1 1;n2 1 @ n1 1;n2 log C B log C ˛: 2 1 @ D @s1 n1 1;n2 D 2 @˛ n1 1;n2 log n1n2 @t1@s1 From these two equations, the logarithmic expression on the right can be elim- 1 B inated, by acting on the first equation with the operator 2 1 .@=@˛/ and on the second with B 1 and subtracting, thus yielding @2 @2 log n1n2 log n1n2 1 @ @t2@s1 @t1@s2 B 1 0 2˛1 B 1 0 ˛1 2 @˛ @2 D @2 log log B n1n2 C B n1n2 C B @t1@s1 C B @t1@s1 C or, equivalently, @ A @ A @2 @2 2 log n1n2 @t2@s1 @t1@s2 B 1 0 1 @2 log B n1n2 C B @t1@s1 C @ @2 A @2 2˛ log n1n2 @ @t2@s1 @t1@s1 0 1 0: (5-10) @˛ @2 D log B n1n2 C B @t1@s1 C Using the remaining Virasoro@ relations (5-9), one obtains along AL the equalities @2 @2 @2 4 log n1n2 F C; 2 2˛ log n1n2 H2C; @t1@s1 D @t2@s1 @t1@s1 ! D @2 @2 @ B 2 2 log n1n2 H1C 2 1 log n1n2 @t2@s1 @t1@s2 ! D @ˇ where we have set1 @ @ B B B B ސ F C 2 1 1 log n1n2 4n1 2 1 1 log n 4n1; WD @˛ D @˛ n Cn n Cn 1 n n 1 2 n1 1 n2 1 n n 1 2 ˛2 One checks that n n .t; s; u ˛;ˇ; ޒ/ L . 2/ 1 2 .2/ 2 j! j! ˛ 1 2 e 2 . 1 2 I j D 0 0 Q Q 386 PIERREVANMOERBEKE @ @ @ B B B B H1C: 0 ˛ ˛ 1 log n1n2 0 1 4 log n1n2 2˛.n1 n2/ D @˛ @˛ C C @˛ C @ @ @ 4n1n2 B0 ˛ ˛B 1 log ސn B0B 1 4 log ސn 4˛n1 D @˛ @˛ C C @˛ C C ˛ @ @ B B B B B H2C: 0 ˛ ˛ 1 log n1n2 .2˛ 1 0 2/ 1log n1n2 2˛.n1 n2/ D @˛ @˛ C C C C @ @ B0 ˛ ˛B 1 log ސn .2˛B 1 B0 2/B 1 log ސn: D @˛ @˛ C C Further define F F C ˛ ˛ ; Hi HiC ˛ ˛ : D n1! n2 D n1! n2 $ $ ˇ ˇ With this notation, equationˇ 5-10 becomes ˇ @ 1 1 B 1 log n n ; F H ; F H ; F G ; 1 2 C 1C 2 C B 1 2C 2 C @=@˛ C @ˇ L B 1 D DW n ˇ o yielding automaticallyˇ a second equation,˚ using« the involu˚ tion ˛ « ˛, ˇ ˇ, ˇ ‘ ‘ n1 n2 (which leaves (5-4) unchanged): $ @ 1 1 B 1 log n n ; F H ; F H ; F G : 1 2 L B 1 2 B 1 2 2 @=@˛ @ˇ 1D DW n ˇ o The last two displaysˇ yield a linear˚ system of equations« ˚ in « ˇ @ log @ log B n1n2 B2 n1n2 1 and 1 @ˇ @ˇ ˇL ˇL from which ˇ ˇ ˇ ˇ ˇ ˇ @ log n1n2 G F C GCF B 1 C ; @ˇ D F .B F / F .B F / ˇL 1 C C 1 ˇ C ˇ B B 2 @ log n1n2 G . 1F C/ GC. 1F / B ˇ C : 1 @ˇ D F .B F / F .B F / ˇL 1 C C 1 ˇ C Subtracting the second equationˇ from B 1 of the first equation yields the dif- ferential equation ˇ F CB 1G F B 1GC F CB 1F F B 1F C C B2 B2 F CG F GC F C 1F F 1F C 0; (5-11) C D which can be rewritten as F C F 0 G G B F B F C F CF det 0 1 C 1 1 0; (5-12) F C C F D B B B B2 B2 1GC 1G C B 1F C 1F C B F C F C @ C A establishing (5-3) for log ސn. ˜ BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 387 6. The Pearcey process As in section 5, consider n 2k nonintersecting Brownian motions on ޒ D (Dyson’s Brownian motions), all starting at the origin, such that the k left paths end up at a and the k right paths end up at a at time t 1. C D Also as observed in section 5, the transition probability can be expressed in terms of the Gaussian Hermitian random matrix probability ސn.˛ E/ with I external source, for which the PDE (5-3) was deduced. Let now the number n 2k of particles go to infinity, and let the points a D and a, properly rescaled, go to . This forces the left k particles to at ˙1 1 t 1 and the right k particles to at t 1. Since the particles all leave from D C1 D the origin at t 0, it is natural to believe that for small times the equilibrium D measure (mean density of particles) is supported by one interval, and for times close to 1, the equilibrium measure is supported by two intervals. With a precise scaling, t 1=2 is critical in the sense that for t < 1=2, the equilibrium measure D for the particles is supported by one, and for t > 1=2, it is supported by two intervals. The Pearcey process P.t/ is now defined [Tracy and Widom 2006] as the motion of an infinite number of nonintersecting Brownian paths, just around time t 1=2 near x 0, with the precise scaling (upon introducing the scaling D D parameter z): 2 1 1 2 n 2k ; a ; xi xiz; t tz ; for z 0: (6-1) D D z4 ˙ D˙z2 ‘ ‘ 2 C ! The Pearcey process has also arisen in the context of various growth models [Okounkov and Reshitikhin 2005]. Even though the pathwise interpretation of P.t/ is unclear and deserves investigation, it is natural to define the following probability for t ޒ, in terms of the probability (5-1), 2 1=z2 1 2 ސ.P.t/ E ?/ lim ސ˙ all xj tz zE 1 j n : \ D WD z 0 0 2 C … I n 2=z4 ! ˇ D ˇ The results of Brezin´ and Hikami [1996; 1997; 1998b; 1998a] for theˇ Pearcey kernel and Tracy and Widom [2006] for the extended kernels show that this limit exists and equals a Fredholm determinant: ސ.P.t/ E ?/ det I Kt ; \ D D E where Kt .x; y/ is the Pearcey kernel, defined as follows: p.x/q00.y/ p0.x/q0.y/ p00.x/q.y/ tp.x/q.y/ Kt .x; y/ C WD x y 1 p.x z/q.y z/ dz; (6-2) D C C Z0 388 PIERREVANMOERBEKE where (note that ! ei=4) D 1 4 2 p.x/ 1 e u =4 tu =2 iuxdu; WD 2 Z 1 1 4 2 q.y/ eu =4 tu =2 uydu WD 2i C ZX ! 4 2 Im 1 du e u =4 .it=2/ u .e!uy e !uy/ D Z0 satisfy the differential equations p tp xp 0 and q tq yq 0: 000 0 D 000 0 C D The contour X is given by the ingoing rays from ei=4 to 0 and the outgoing ˙1 rays from 0 to e i=4, i.e., X stands for the contour ˙1 -0 . % & r ޒ For compact E i 1Œx2i 1; x2i , define the gradient and the Euler D operator with regard toD the boundary points of E, S 2r 2r @ @ B 1 ; B0 xi : (6-3) D @xi D @xi X1 X1 THEOREM 6.1 [Adler and van Moerbeke 2007]. Q.t x1;:::; x2r / log ސ P.t/ E ? log det .I Kt E/ (6-4) I WD \ D D satisfies a fourth-order, third-degree PDE, which can be written as a single Wronskian: 1 @3Q 1 @Q @Q .B 2/B2 Q B ; B2 Q ; B2 0: 3 0 1 1 1 1 2 16 @t B 1 @t ( @t C C )B D n o 1 (6-5) REMARK. A similar PDE can be written for the transition probability involving several times; see [Adler and van Moerbeke 2006]. Such equations can be used to compute the asymptotic behavior of the Pearcey process for t . ! 1 SKETCH OF PROOF. Consider the function Qz.s x1;:::; x2r /, defined in terms ސ a ސ I of the probabilities 0˙ , defined in (5-1) and n, defined in (5-2), as follows: ސ a Qz.s x1;:::; x2r / log 0˙ .t b1;:::; b2r / n 2=z4; a 1=z2; I WD I D D1 2 bi xi z; t sz ˇ D D 2 C ˇ BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 389 2t 2 2 log ސn a b1 ; :::; b2r 4 2 D 1 t I t.1 t/ t.1 t/ n 2=z ; a 1=z ; r r r ˇ b D x z; t D1 sz2 ˇ i i 2 ˇ D D C 1 2 ˇ p2 2 sz x1zp2 x2r zp2 log ސ2=z4 C ;:::; ; D z2 v 1 2 I 1 1 u 2 sz s2z4 s2z4 ! u 4 4 t q q from which it follows, by inversion, that 2 4 u z 2 v1uz v2r uz Qz ;:::; 2z2 u2z4 2 I u2z4 2 u2z4 2 C C C ! log ސ 4 .u v1;:::;v2r /: (6-6) D 2=z I This expression satisfies the PDE (5-3), with ˛ and b1;:::; b2r replaced by u and v1;:::;v2r . Therefore all the partials of log ސ with regard to these variables u and v1;:::;vr , as appears in the PDE (5-3), can be expressed, by virtue of (6-6), by partials of Qz with regard to s and x1;:::; x2r . For this, we need to compute the expressions F ; B F ; B2 F ; G and ˙ Q 1 ˙ Q 1 ˙ ˙ B Q 1G˙ appearing in (5-3) (where we use tildes in contrast to the operators defined in (6-3)), in terms of Qz.s x1;:::; x2r / I 1 2 p2 2 sz zp2 zp2 log ސ2=z4 C x1 ;:::; x2r D z2 v 1 2 I 1 1 u 2 sz s2z4 s2z4 ! u 4 4 t Q.s x1;:::; x2r / O.z/; q q (6-7) D I C with Q.s x1;:::; x2r / log det I Ks c : (6-8) I D E Without taking the limit z 0 on ޑz.s x1;:::; x2r / yet, one computes, upon ! I setting " , WD ˙ 4 1 " @Q F " B2 Q B z O.z/; 4 2 1 z 1 D z 4z C 4z @s C 1 1 " @Q "s @Q B F " B3 Q B2 z B2 z O.z/; 1 3 1 z 2 1 1 p2 Q D 16z C 16z @s 8 @s C 1 " @Q "s @Q B2 F " B4 Q B3 z B3 z O.1/; 1 4 1 z 3 1 1 Q D 32z C 32z @s 16z @s C 390 PIERREVANMOERBEKE 3" "s G" B3 Q B3 Q 9 1 z 7 1 z D 8z C 4z 1 @Q B z .B3 Q / 32B B2 Q 6 1 1 z 0 1 z 128z @s C 3 @Qz @ Qz 1 .B2 Q 64s/B2 64B2 Q 16 O ; 1 z 1 1 z 3 5 C @s C @s C z 1 3" "s B G" B4 Q B4 Q 1 10 1 z 8 1 z p2 Q D 32z C 16z 1 @Q B z B4 B B3 1 . 1Qz/ 32 0 1Qz C 512z7 @s @Q B2 B3 z . 1Q 64s/ 1 C C @s 3 @ Qz 1 32B3 Q 16B O : 1 z 1 3 6 C @s C z Using these expressions, one easily deduces for small z, 0 F CB 1G F B 1GC F CB 1F F B 1F C D Q C Q Q Q B2 B2 F CG F GC F C 1F F 1F C C Q Q " @Q 1 @3Q B2 z ; z .B 2/B2 Q 17 1 3 0 1 z D 2z @s 2 @s C B 1 n @Q o @Q 1 B z B3 B2 z 1 1 1Qz; 1 O 15 C 16 @s @s B 1 C z n o " 1 the same expression for Q.s x1;:::; x2r / O ; D 2z17 I C z16 using (6-8) in the last equality. Taking the limit when z 0 yields equation 6-5 ! of Theorem 6.1. ˜ 7. The Airy process Consider n nonintersecting Brownian motions on ޒ, all leaving from the ori- gin and forced to return to the origin. According to formula (4-2), this proba- bility, 0 ˘ ސ all xi.t/ E all xj .0/ xj .1/ 0 ; WD 0 2 D D can be expressed in terms of the determinantˇ of a moment matri x and further ˇ as an integral over Hermitian matrices, both with rescaled space, for 0 t 1. To do this we let Hn.E/ denote the space of n n Hermitian matrices with BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 391 spectrum in the set E ޒ, and one checks that 1 i j y2=2 ˘ det dy y C e D Zn E.p2=pt.1 t// 0 i;j n 1 Z 1 Tr M 2 e dM: D Zn0 Hn.E.1=pt.1 t/// Z The Airy process A. / describes the nonintersecting Brownian motions above for large n, but viewed from the (right-hand) edge 2nt.1 t/ of the set of particles, with time and space properly rescaled, so that the new time scale p equals 0 when t 1=2. Random matrix theory suggests the following time and D space rescaling (edge rescaling): . ; x/ p2n 1 1 C p2n1=6 t ; E : 1=3 D 1 e 2 =n D 2 cosh C n1=3 Taking the limit when n , one finds that the rescaled motion becomes ! 1 time-independent (stationary), P.A. / x/ . ; x/ p2n 1 1=6 0 1 C p2n lim ސ all xi all xj .0/ xj .1/ 0 WD n 0 1 e 2 =n1=3 2 2 cosh. =n1=3/ D D !1 ˇ C ˇ 1 ˇ Tr M 2 lim ˇ e dM n 1=6 D Zn Hn p2n .. ;x/=p2n / !1 Z C 1 x lim Prob .all eigenvalues of M / p2n D n C p2n1=6 !1 exp 1.˛ x/g2.˛/d˛ D Zx F2.x/ Tracy–Widom distribution; DW D with g.˛/ the unique solution of g ˛g 2g3 00 D C .2=3/ ˛3=2 .Painleve´ II/: (7-1) 8 e ˆg.˛/ for ˛ : < Š 2p˛1=4 % 1 This is to say:ˆ the outmost particle in the nonintersecting Brownian motions fluctuates according to the Tracy–Widom distribution [1994] for n . ! 1 Since the Airy process is stationary, the joint distribution for two times t1 < t2 in Œ0; 1 is of interest; here one checks that 392 PIERREVANMOERBEKE 0 ސ all xi.t1/ E1; all xi.t2/ E2 all xj .0/ xj .1/ 0 0 2 2 D D ˇ t1.1 t2/ ˇ 2t2 2.1 t1/ ސn E1 ; E2 ; (7-2) D st2.1 t1/I .t2 t1/t1 s.1 t2/.t2 t1/ s ! where 1 1 2 2 ސ 2 Tr.M1 M2 2cM1M2/ n.c E10 ; E20 / dM1 dM2 e C I WD Zn H.E0 / H.E0 / ZZ 1 2 N 1 2 2 2 .xk yk 2cxk yk / cN0 N .x/N .y/ e C dxk dyk : D EN ZZ k 1 YD According to [Adler and van Moerbeke 1999b], given r s 2 E E1 E2 Œa2i 1; a2i Œb2i 1; b2i ޒ ; (7-3) D WD i 1 i 1 SD SD log ސn.c E1; E2/ satisfies a nonlinear third-order partial differential equation I (in terms of the Wronskian f; g X g.Xf/ f.Xg/, with regard to the first f g D order differential operator X ): nc B A log ސ ; B A log ސ 2 1 n 1 1 n 2 C c 1 A1 n o nc A B log ސ ; A B log ސ 0: (7-4) 2 1 n 1 1 n 2 C c 1 B1 D n o in terms of the differential operators, depending on the coupling term c and the boundary of E, r s r s @ @ @ @ A 1 B 1 1 2 c ; 1 2 c ; D c 1 @aj C @bj D 1 c @aj C @bj 1 1 1 1 r X X s X X @ @ @ @ A B 2 aj c ; 2 bj c : (7-5) D @aj @c D @bj @c j 1 j 1 XD XD Using the same rescaled space and time variables, as before, introduce new times 1 < 2 and points x; y ޒ, defined as 2 . ; x/ . ; y/ p2n 1 p2n 1 1 C p2n1=6 C p2n1=6 t ; E ; E : i 1=3 1 2 D 1 e 2 i =n D 2 cosh 1 D 2 cosh 2 C n1=3 n1=3 BROWNIAN MOTIONS, INTEGRABLE SYSTEMS AND ORTHOGONAL POLYNOMIALS 393 One verifies, in view of (7-2), that p p . ; x/ 2 2n 11=6 2t2 C p2n E1 ; 1=3 s.t2 t1/t1 D 1 e 2. 2 1/=n p p p . ; y/ 2 2n 11=6 2.1 t1/ C p2n E2 ; 1=3 s.1 t2/.t2 t1/ D 1 e 2. 2 1/=n p t .1 t / 1=3 1 2 . 2 1/=n c e : D st2.1 t1/ D Defining ޑ. 2 1 x; y/ I WD x y 2pn 2pn 1=3 1=6 1=6 . 2 1/=n C n C n log ސn e ; ; 1=3 1=3 I 1 e 2. 2 1/=n 1 e 2. 2 1/=n ! one checks, setting z n 1=6 andp using the inverse map,p that D 2 1 2 4 1 2 4 log ސn.c a; b/ ޑ z log c az 1 c 2z ; bz 1 c 2z : I D I p p But log ސn.c E1; E 2/ satisfies the PDE (7-4), which induces a PDE for ޑ; then I letting z , the leading term in this series must be 0. One finds thus the ! 1 D following PDE for the Airy joint probability, namely H.t x; y/ log P .A. 1/ y x; A. 2/ y x/; I WD C 2 takes on the following simple form in x; y and t , with t 2 1, also involving D a Wronskian (see [Adler and van Moerbeke 2005]) @3H t 2 @ @ @2H @2H @2H @2H 2t x ; ; (7-6) @t@x@y D 2 @x @y @x2 @y2 C @x@y @y2 y with initial condition lim H .t x; y/ log F2 .min.y x; y x// : t 0 I D C & The edge sup A.t/ of the cloud is non-Markovian, as is the largest particle in the finite nonintersecting Brownian problem. As t 2 1 , the edges D ! 1 sup A. 1/ and sup A. 2/ become independent. This poses the question: How 394 PIERREVANMOERBEKE much does the process remember from the remote past? The following asymp- totics for the covariance of the edge of the cloud, for large t 2 1, is deduced D from the PDE: E.sup A. 2/ sup A. 1// E.sup A. 2//E.sup A. 1// 1 2 ˚.u;v/ du dv ; 2 4 D t C t ޒ2 C ZZ where 2 2 1 1 2 1 2 ˚.u;v/ F2.u/F2.v/ g d˛ g d˛ WD 4 Zu Zv 1 1 2 g2.u/ g2.v/ 1g2d˛ C 4 2 Zv 1d˛ 2.v ˛/g2 g 2 g4 1g2d˛ : C C 0 Zv Zu (Here g g.˛/ is the function (7-1) and F2.u/ is the Tracy–Widom distribu- D tion.) The Airy process was introduced by Spohn and Prahofer¨ [2002] in the context of polynuclear growth models. It has been further investigated by Johansson [2001; 2003; 2005], by Tracy and Widom [2004] and by Adler and van Moer- beke [2005]; see also [Widom 2004]. References [Adler and van Moerbeke 1997] M. Adler and P. van Moerbeke, “String-orthogonal polynomials, string equations, and 2-Toda symmetries”, Comm. Pure Appl. Math. 50:3 (1997), 241–290. Available at arxiv.org/abs/hep-th/9706182. [Adler and van Moerbeke 1999a] M. 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PIERRE VAN MOERBEKE DEPARTMENT OF MATHEMATICS UNIVERSITE´ CATHOLIQUE DE LOUVAIN 1348 LOUVAIN-LA-NEUVE BELGIUM and BRANDEIS UNIVERSITY WALTHAM, MA 02454 UNITED STATES [email protected]