AU0121208 Application of the r-function theory of Painleve equations to random matrices: Pv , Pin , the LUE, JUE and CUE P.J. Forrester and N.S. Witte* Department of Mathematics and Statistics ^(and School of Physics), University of Melbourne, Victoria 3010, Australia ; email: [email protected]; [email protected] With (•) denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is £W(/;a,£i) := (IIili X(o oo)\/(^ ~ A;)M) f°r I = (0>s) and I = (s;°°)i where xp = 1 for A; € / and xj — 0 otherwise. Using Okamoto's development of the theory of the Painleve V equation, it is shown that EN{I;CL,[I,) is a r-function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second order second degree differential equation. The cases fj, = 0 and \i = 2 are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case / = (s,oo), EN{I\CL,H) is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of £V(/;a,^). In particular, in the hard edge scaled limit it is shown that the limiting quantity £hard((0, s);a,fi) can be evaluated as a r-function associated with the Hamiltonian in Okamoto's theory of the Painleve III equation. keywords: random matrices - Painleve equations - Toda lattice equation - Backlund transformations - root systems - affine Weyl groups - classical orthogonal polynomial systems 1 Introduction and summary In a previous paper [17] the quantities and (n) (1.2) where the averages are with respect to the joint eigenvalue distribution of the Gaussian unitary ensemble (GUE), were shown to be equal to the r-functions occurring in Okamoto's theory [28] of the Painleve IV equation. It was noted in [17] that we expect the analogous quantities for the Laguerre unitary ensemble (LUE) to be expressible in terms of the -r-functions occurring in Okamoto's theory [29] of the Painleve V equation. It is the purpose of this article to verify this statement by giving the details of the correspondences between the multi-dimensional integrals defining the analogues of JSAT(A;O) and FN(\;O) for the LUE, and the r-functions from [29]. Let us first recall the definition of the LUE. Let X be a n x N (n > N) Gaussian random matrix of complex elements Zjk, with each element independent and distributed according to the Gaussian density ^e"'2-**' so that the joint density of X is proportional to e-Mtx. (1.3) Denote by A the non-negative matrix X'X. Because (1.3) is unchanged by the replacement X i-> UXV for U a n x n unitary matrix and V a JV x JV unitary matrix, the ensemble of matrices A is said to have a unitary 32/ 40 symmetry. The probability density function (PDF) for the eigenvalues of A is given by 1 N c n 1=1 where C denotes the normalization and a = n — JV, n > N. (Throughout, unless otherwise stated, the symbol C will be used to denote some constant i.e. a quantity independent of the primary variables of the equation.) Because Aae~* is the weight function occurring in the theory of the Laguerre orthogonal polynomials, and the ensemble of matrices A has the aforementioned unitary symmetry, the eigenvalue PDF (1.4) is said to define the LUE. Analogous to EN(X;O) specified by (1.1) for the GUE, we introduce the quantities )LUE EN ((a, oo); a, p) := ( f[f 5$ (s -A,)") (1.6) where the averages are with respect to (1.4) (the parameter a in (1.1) has been denoted /J, in (1.5), (1.6) to avoid confusion with the parameter a in (1.4)). Explicitly EN((0,s);a,fi) 1 /*oo /"oo •*• / /-7\ \^rt — 1 (\ \iii I i-7\ \^ — N (\ "^^ T F (\ \ \^ ^ \ f „ \ 1—r o / TT (A* " \<j<k<N S 1 a A dAl A-(i _ Xlye- ^ ••• f d\N X N(l - A* )"e"' " IT In the case /i = 0 the first integrals in (1.7) and (1.8) are the definitions of the probability that there are no eigenvalues in the intervals (0, s) and (s, oo) respectively of the Laguerre unitary ensemble. The case p, — 2 also has significance in this context. To see this, first note from the definitions (using the first integral representation in each case) that a ^-EN+1 ((0, a); a, 0) oc s e—EN((0, s); a, 2) (1.9) a s •fEN+1{(s,oo);a,0) oc 8 e- EN((s,oo);a,2). (1.10) CIS On the other hand, with pmin(s; a) and pm&x(s; a) denoting the distribution of the smallest and largest eigenvalue respectively in the N x N LUE, we have pTnin(s;a) = —^EN{{O,s);a,O) (1.11) as Pmax(s;a) = -T- EN((s,oo);a,0). (1.12) Hence for JV i-> JV+1, pmin(s;o) andpmax(s; o) are determined by EN((0, S); a, 2) and J5AT((S,OO);O, 2) respectively. From the second formula in (1.7), we see that with N ) (1-13) 'LUE we have Ns FN(s;a,fJ,)= (e EN((O,s);n,a))\ (1.14) (notice the dual role played by fi and a on the different sides of (1.14)) so there is no need to consider FM separately. Note that with /j, = 2, (1.13) multiplied by sae~s is proportional to the definition of the density in the LUE with N *-+ N + 1. The second integral in (1.8) is of interest for its relevance to the Jacobi unitary ensemble, which is specified by the eigenvalue PDF N This ensemble is realized by matrices of the form A(A + B)"1, where A = X^X, B = Y^Y, for X (Y) annixJV (n2 x N) complex Gaussian random matrix with joint density (1.3). The parameters o and 6 are then specified by a = ni — N, b — ri2 — N (c.f. the value of o in (1.4)). We see from (1.15) that 1 A a b sX 2 s ~ f dAi Af (1 - AX)V > • • • / d\N \ N(l - \N) e » TT (Afc - A,) = (e ^=^ *') , (1.16) CJ° Jo i<f<t<N X /juE so substituting in (1.8) and compensating for the different normalizations in (1.8) and (1.15) shows JUE where Xl a XN X IN[a):= [ d\i\1e- ••• [ d\N \ Ne~ ]J ( k - Xjf (118) 2 JN(a,fj,):= f dAiA;(l-Ai)""- / dA^A^a-Aivf J][ (\k-\j) . (1.19) Jo Jo l<j<k<N As is well known, the integrals IN (a) and Jjv(a,/u) can be evaluated in the form Af!n=o CI> where Cj is the normalization of the monic orthogonal polynomial of degree j associated with the weight functions Aae~A and Aa(l-A)'i respectively. One important feature of the JUE is that with the change of variables (cos^ + 1), (1.20) the PDF (1.15) assumes a trigonometric form, which for appropriate (TV, a, 6) coincides with the PDF for the independent eigenvalues e10' of random orthogonal and random unitary symplectic matrices. In the case of orthogonal matrices, one must distinguish the two classes 0% and O^ according to the determinant equalling +1 or —1 respectively. The cases of ./V even and N odd must also be distinguished. For N odd all but one eigenvalue come in complex conjugate pairs e±t9j, with the remaining eigenvalue equalling +1 for O% and —1 for OJ,. For N even, all eigenvalues of O~^ come in complex conjugate pairs, while for Ojf all but two eigenvalues come in complex conjugate pairs, with the remaining two equalling ±1. Let us replace N in (1.15) by N* and make the change of variables (1.20). Then the PDF for the independent eigenvalues of an ensemble of random orthogonal matrices is (see e.g. [10]) of the form (1.15) with (JV/2,-1/2,-1/2) for matrices in O£, iVeven NN 22 (N* b) = J dd ~~ i)/i)/ ' ' -1/2,1/2-1/2,1/2)) foforr matricematricess iinn O+,O+, AATT od oddd 1 'a' > > ((JV-l)/2,1/2,-1/2((JV-l)/2,1/2,-1/2)) foforr matricematrices iin O~O~,, AAT od oddd ; (JV/2 - 1,1/2,1/2) for matrices in O^, JVeven Matrices in the group USp(N) are equivalent to 2JV x 2N unitary matrices in which each 2x2 block has a real quaternion structure. The eigenvalues come in complex conjugate pairs e », and the PDF of the independent elements is of the form (1.15) with the change of variables (1.20) and {N\a,b) = {N,1/2,1/2). (1.22) It follows from the above revision that (esT«u)) (1.23) for G = O%,O^ or USp(N) is a special case of the more general average (1.16). Explicitly, T y / < >\ ""/2^x\ (124) uea \ /JUE where on the RHS the dimension of the JUE is at first N*, then the parameters (JV*, a, b) are specified as in (1.21) or (1.22).