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Hermitian, Symmetric and Symplectic Random Ensembles Annals of Mathematics, 153 (2001), 149–189 Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum By M. Adler and P. van Moerbeke* Abstract Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals sat- isfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary points of the intervals) is related to the Toda lattice and the KP equation, whereas for the symmetric and symplectic ensembles the PDE is an inductive equation, related to the so-called Pfaff-KP equation and the Pfaff lattice. The method consists of inserting time-variables in the integral and showing that this integral satisfies integrable lattice equations and Virasoro constraints. Contents 0. Introduction 0.1. Hermitian, symmetric and symplectic Gaussian ensembles 0.2. Hermitian, symmetric and symplectic Laguerre ensembles 0.3. Hermitian, symmetric and symplectic Jacobi ensembles 0.4. ODEs, when E has one boundary point 1. Beta-integrals 1.1. Virasoro constraints for β-integrals 1.2. Proof: β-integrals as fixed points of vertex operators 1.3. Examples arXiv:math-ph/0009001v2 14 Aug 2001 2. Matrix integrals and associated integrable systems 2.1. Hermitian matrix integrals and the Toda lattice 2.2. Symmetric/symplectic matrix integrals and the Pfaff lattice 3. Expressing t-partials in terms of boundary-partials 3.1. Gaussian and Laguerre ensembles 3.2. Jacobi ensemble 3.3. Evaluating the matrix integrals on the full range ∗The support of a National Science Foundation grant #DMS-98-4-50790 is gratefully acknowl- edged. The support of a National Science Foundation grant #DMS-98-4-50790, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. 150 M. ADLER AND P. VAN MOERBEKE 4. Proof of Theorems 0.1, 0.2, 0.3 4.1. β = 2, 1 4.2. β = 4, using duality 4.3. Reduction to Chazy and Painlev´eequations (β = 2) 5. Appendix. Self-similarity proof of the Virasoro constraints (Theorem 1.1) 0. Introduction V (z) Consider weights of the form ρ(z)dz := e− dz on an interval F = [A, B] R, with rational logarithmic derivative and subjected to the following ⊆ boundary conditions: i ρ′ g 0∞ biz k (0.0.1) = V ′ = = i , lim f(z)ρ(z)z = 0 for all k 0, − ρ f ∞ a z z A,B ≥ P0 i → together with a disjoint unionP of intervals, r (0.0.2) E = [c2i 1, c2i] F R. − ⊆ ⊆ [1 The data (0.0.1) and (0.0.2) define an algebra of differential operators 2r k+1 ∂ (0.0.3) k = ci f(ci) . B ∂ci X1 Let n, n and n denote the Hermitian (M = M¯ ⊤), symmetric (M = M ⊤) H S T 1 and “symplectic” ensembles (M = M¯ ⊤, M = JMJ¯ − ), respectively. Tra- ditionally, the latter is called the “symplectic ensemble,” although the ma- trices involved are not symplectic! These conditions guarantee the reality of the spectrum of M. Then, (E), (E) and (E) denote the subsets of Hn Sn Tn , and with spectrum in the subset E F R. The aim of this paper Hn Sn Tn ⊆ ⊆ is to find PDEs for the probabilities (0.0.4) P (E): = P ( all spectral points of M E) n n ∈ e tr V (M)dM n(E), n(E) or n(E) − = H S T e tr V (M)dM R n(F ), n(F ) or n(F ) − H S T R β n V (zk) En ∆n(z) k=1 e− dzk = | |β n V (z ) , β = 2, 1, 4 respectively, n ∆ (z) e k dz RF | n | Qk=1 − k for the Gaussian,R Laguerre andQ Jacobi weights. The probabilities involve pa- rameters β, a, b (see (0.1.1), (0.2.1) and (0.3.2)) and 1/2 1/2 2 β β − 0 for β = 2 δβ := 2 = 1,4 2 2 1 for β = 1, 4. − ! ( RANDOM ENSEMBLES 151 The method used to obtain these PDEs involves inserting time-parameters into the integrals, appearing in (0.0.4) and to notice that the integrals obtained satisfy Virasoro constraints: linear PDEs in t and the boundary points of E, and • integrable hierarchies: • ensemble β lattice Hermitian β = 2 Toda symmetric β = 1 Pfaff symplectic β = 4 Pfaff . As a consequence of a duality (explained in Theorem 1.1) between β-Virasoro generators under the map β 4/β, the PDEs obtained have a remarkable 7→ property: the coefficients Q and Qi in the PDEs are functions of the variables n, β, a, b, and have the invariance property under the map a b n 2n, a , b ; →− →−2 →−2 to be precise, a b (0.0.5) Qi( 2n,β, , ) = Qi(n, β, a, b) β=4 . − −2 −2 β=1 | Important remark. For β = 2, the probabilities satisfy PDEs in the bound- ary points of E, whereas in the case β = 1, 4, the equations are inductive. Namely, for β = 1 (resp. β = 4), the probabilities Pn+2 (resp. Pn+1) are given in terms of Pn 2 (resp. Pn 1) and a differential operator acting on Pn. − − 0.1. Hermitian, symmetric and symplectic Gaussian ensembles. Given the 2 disjoint union E R and the weight e bz , the differential operators take ⊂ − Bk on the form 2r k+1 ∂ k = ci . B ∂ci X1 Also, define the invariant polynomials (in the sense of (0.0.5)) 2 2 Q = 12b2n n + 1 , Q = 4(1 + δβ )b 2n + δβ (1 ) β 2 1,4 1,4 β − − and b2 Q = 2 δβ . 1 − 1,4 β Theorem 0.1. The following probabilities for (β = 2, 1, 4) 2 β n bzk En ∆n(z) k=1 e− dzk (0.1.1) Pn(E)= | | , β n bz2 RRn ∆n(z) e− k dzk | | Qk=1 R Q 152 M. ADLER AND P. VAN MOERBEKE satisfy the PDE’s (F := Fn = log Pn): (0.1.2) Pn 2 Pn+ 2 2 when n is even and β = 1 β − 1 1 with index δ1,4Q 2 1 Pn − ! ( 1 when n is arbitrary and β = 4 4 2 2 2 = 1 + (Q2 + 6 1F ) 1 + 4Q1(3 0 4 1 1 + 6 0) F. B− B− B− B − B− B B 0.2. Hermitian, symmetric and symplectic Laguerre ensembles. Given the disjoint union E R+ and the weight zae bz, the take on the form ⊂ − Bk 2r k+2 ∂ k = ci . B ∂ci X1 Also define the polynomials, again respecting the duality (0.0.5), 3 n(n 1)(n + 2a)(n + 2a + 1), for β = 1 4 − Q = , 3 n(2n + 1)(2n + a)(2n + a 1), for β = 4 2 − 2 2 a β β 2 β Q2 = 3βn + 6an + 4(1 )a + 3 δ1,4 + (1 a )(1 δ1,4), − β − 2 ! − − β a Q = βn2 + 2an + (1 )a , Q = b(2 δβ )(n + ), 1 2 0 1,4 β − − 2 b β Q 1 = 2 δ1,4 . − β − Theorem 0.2. The following probabilities β n a bzk En ∆n(z) k=1 zk e− dzk (0.2.1) Pn(E)= | | β n a bzk Rn ∆n(z) k=1 zk e− dzk R + | | Q 1 R Q satisfy the PDE : (F := Fn = log Pn) (0.2.2) Pn 2 Pn+ 2 β − 1 1 δ1,4Q 2 1 Pn − ! 4 β 3 = 1 2(δ1,4 + 1) 1 B− − B− 2 β 2 β + (Q2 + 6 1F 4(δ1,4 + 1) 1F ) 1 3δ1,4(Q1 1F ) 1 B− − B− B− − −B− B− 2 + Q 1(3 0 4 1 1 2 1)+ Q0(2 0 1 0) F. − B − B B− − B B B− −B 1with the same convention on the indices n 2 and n 1, as in (0.1.2) ± ± RANDOM ENSEMBLES 153 0.3. Hermitian, symmetric and symplectic Jacobi ensembles. In terms of E [ 1, 1] and the Jacobi weight (1 z)a(1 + z)b, the differential operators ⊂ − − k take on the form B 2r k+1 2 ∂ k = ci (1 ci ) . B − ∂ci X1 Setting b = a b, b = a+b, we introduce the new variables, which themselves 0 − 1 have the invariance property (0.0.5): 4 4 r = (b2 + (b + 2 β)2) s = b (b + 2 β) β 0 1 − β 0 1 − 4 q = (βn + b + 2 β)(βn + b ), n β 1 − 1 and the following polynomials in q = qn,r,s, thus invariant under the map (0.0.5): 3 (0.3.1) Q = (s2 qr + q2)2 4(rs2 4qs2 4s2 + q2r) , 16 − − − − Q = 3s2 3qr 6r + 2q2 + 23q + 24, 1 − − Q = 3qs2 + 9s2 4q2 r + 2qr + 4q3 + 10q2, 2 − Q = 3qs2 + 6s2 3q2r + q3 + 4q2, 3 − Q = 9s2 3qr 6 r + q2 + 22q +24 = Q + (6s2 q2 q). 4 − − 1 − − Theorem 0.3. The following probabilities β n a b En ∆n(z) k=1(1 zk) (1 + zk) dzk (0.3.2) Pn(E)= | | β n − a b [R 1,1]n ∆n(z) Q k=1(1 zk) (1 + zk) dzk − | | − R Q satisfy the PDE (F = Fn = log Pn): for β = 2: (0.3.3) 4 2 2 2 2 1 + (q r + 4) 1 (4 1F s) 1 + 3q 0 2q 0 + 8 0 1 B− − B− − B− − B− B − B B B− 4(q 1) 1 1 + (4 1F s) 1 + 2(4 1F s) 0 1 + 2q 2 F − − B B− B− − B B− − B B− B 2 2 +4 1F 2 0F + 3 1F = 0 B− B B− 154 M.
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