Elsevier

Nuclear Physics B

Volume 743, Issue 3, 29 May 2006, Pages 307-332
Nuclear Physics B

Hermite and Laguerre β-ensembles: Asymptotic corrections to the eigenvalue density

https://doi.org/10.1016/j.nuclphysb.2006.03.002Get rights and content

Abstract

We consider Hermite and Laguerre β-ensembles of large N×N random matrices. For all β even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polynomials. The corrections to the bulk density are oscillatory terms that depends on β. At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes a β-deformation of the Airy function. This allows us to obtain the main contribution to the soft edge density when the spectral parameter tends to ±∞.

Introduction

We deal with two families of N×N random matrices: the Hermite and Laguerre β-ensembles (for a review see [9]). These ensembles possess an eigenvalue joint probability density function (p.d.f.) of the formPN,β(x)=1ZNeβW(x),x=(x1,,xN)IN, where β is real and positive. The support I of the eigenvalues in the Hermite and Laguerre cases are respectively (,) and (0,). The ensembles' names come from the fact that their p.d.f. generalize the weight functions related to the Hermite and Laguerre polynomials; that is,W(x)={12i=1Nxi21i<jNln|xixj|,Hermite,12i=1Nxia2i=1Nln|xi|1i<jNln|xixj|,Laguerre, where a is a real and non-negative parameter. The normalization constants can be computed with the help of the Selberg integrals:ZN={Gβ,N:=gβ,Nj=2NΓ(1+jβ/2)Γ(1+β/2),Hermite,Wa,β,N:=wa,β,Nj=1NΓ(1+jβ/2)Γ(1+(a+j1)β/2)Γ(1+β/2),Laguerre, where gβ,N=(2π)N/2βN(1/2+β(N1)/4) and wa,β,N=(2/β)N(aβ/2+1+β(N1)/2).

For special values of the Dyson index β, we recover classical random matrix ensembles (see e.g. [9], [19]). Indeed, the β=1,2, and 4 Hermite ensembles are respectively equivalent to the Gaussian orthogonal, unitary, and symplectic ensembles. The Laguerre ensembles are similarly related to the real, complex and quaternionic Wishart matrices. Recently, Dumitriu and Edelman [3] have constructed explicit random matrices associated to the Hermite and Laguerre p.d.f. given in Eq. (1). A generic random N×N matrix belonging to the Hermite β-ensemble can be written as a tridiagonal symmetric matrix:Hβ=1β(N[0,1]χ(N1)βχ(N1)βN[0,1]χ(N2)βχ(N2)βN[0,1]χ(N3)βχ2βN[0,1]χβχβN[0,1]). This means that the N diagonal elements and the N1 subdiagonal elements are mutually independent; the diagonal elements are normally distributed (with mean zero and variance 1) while the off-diagonal have a chi distribution. Recall that the densities associated to N[μ,σ] and χk are respectively (2πσ2)1/2e(xμ)2/(2σ2) and 2xk1ex2/Γ(k/2), where in the latter case x>0. Any N×N matrix Lβ of the Laguerre β-ensemble also has a tridiagonal form: Lβ=BβTBβ, for some N×N matrixBβ=1β(χPβχ(N1)βχ(P1)βχ(N2)βχ(PN+1)βχβ),a=PN+12β.

In this article, we compute the density for large but finite random matrices of the Hermite and Laguerre β-ensembles. The density, or the marginal eigenvalue probability density, is defined as follows:ρN,β(x):=NZNINPN,β(x1,,xN)dx1dxN. The quantity N−1ρN,β(x)dx represents the probability to have an eigenvalue in the interval [x,x+dx]. The density has two simple physical interpretations.

First, we remark that the Hermite p.d.f. is equivalent to the Boltzmann factor of a log-potential Coulomb gas with particles of charge unity confined to the interval (2N,2N) with neutralizing background charge density (2N/π)1x2/2N. From this point of view, ZN (divided by N!) is simply the canonical partition function at inverse temperature β and ρN,β(x)dx gives the number of charges present in the interval [x,x+dx]. This analogy allows one to predict the global density:limN2NρN,β(2Nx)=ρW(x):={2π1x2,1<x<1,0,|x|1. This result is known as the Wigner semicircle law. For a finite matrix, we expect that the scaled density is of order one in the interval (2N,2N), the ‘bulk region’ of the mechanical problem, while it decreases rapidly around ±2N, called the ‘soft edges’. A similar log-gas construction is possible for the Laguerre case. One expects the c=1 Marčenko–Pastur law [18]:limN4ρN,β(4Nx)=ρMP(x):={2π1x1,0<x<1,0,x1. We see that, in the Laguerre case, the ‘bulk’ is (0,4N) while the ‘soft edge’ is the point 4N. The origin is referred as the ‘hard edge’ of the support because the eigenvalues are constrained to be positive. The predictions given in Eqs. (5), (6) have been confirmed in [1], [7]. The asymptotic analysis used in these references constitutes the starting point for the study of the higher expansions to be undertaken in the present work.

Second, their is a deep connection between the β-ensembles and some integrable quantum mechanical N-body problems on the line, known as the Calogero–Moser–Sutherland (CMS) models (a good reference is [21]). The Hermite p.d.f. is in fact the ground state wave functions squared of the (rational) AN1 CMS model, whose Hamiltonian isH(H)=i=1N2xi2+β24i=1Nxi2+β(β2)21i<jN1(xixj)2, for xj(,). The Laguerre p.d.f. is the ground state squared of the Hamiltonian of the BN CMS model, which can be expressed as follows:H(L)=2i=1N(2xi2xi2+xi)+14i=1N(aβ(aβ2)1xi+β2xi)+β(β2)1i<jNxi+xj(xixj)2, where xj(0,). It has been shown in [1] (see also [23]) that the eigenfunctions of the conjugated Schrödinger operators eβW/2H(H)eβW/2 and eβW/2H(L)eβW/2 are respectively the generalized (or multivariate) Hermite and Laguerre polynomials, previously introduced by Lassalle in [16], [17]. In the context of CMS models, the global density can be seen as the ground state expectation value of the density operator ρˆ(x)=j=1,,Nδ(xxj), also known as the one-point function.

The relation between the CMS models and the generalized classical orthogonal polynomials furnishes, when β is an even integer, new integral representations of the global density that suits perfectly for asymptotic analysis. Let us be more explicit. The definition of the density given in (4) contains N integrals; considering N large does not simplify the calculation. On the other hand, it has been noticed in [1], [7] that the density is a particular Hermite (or Laguerre) polynomial, characterized by a partition λ=((N1)β) and evaluated at x1==xβ=x (see below). Using the work of Kaneko [14] and Yan [22], one then can shows that the density is proportional to the following β-dimensional integral:RN,β(x):=Cdu1eNf(u1,x)CduβeNf(uβ,x)1j<kβ|ujuk|4/β, for a particular contour C and function f(u,x).

In the following sections, we apply the steepest descent method [20], [24] to integrals of the type (7). We obtain expressions for the density in the bulk and at the soft edge that are valid for every β2N. Of course, these results generalize many known result obtained for β=2 and 4. We mention in particular two recent publications in which asymptotic corrections to the global density have been obtained: (1) Kalisch and Braak [13] for some ensembles, including the Gaussian unitary and symplectic ensembles (work based on the supersymmetric method); (2) Garoni et al. [11] for the Laguerre and Gaussian unitary ensembles (calculations using the theory of orthogonal polynomials). Also, the preprint [10] of Forrester, Frankel and Garoni addresses the Laguerre and Gaussian ensembles with orthogonal and symplectic symmetry. All studies show that these approximate expressions of the global density are very accurate, even for N=10, say (for instance, see Figs. 1 and 2 in [11]). We finally point out that an asymptotic formula for the density in the Hermite β-ensemble has been considered in a different context: Johansson [12] has studied a smoothed (macroscopic) density and has derived corrections of order 1/N to Eq. (5). However, contrary to the asymptotic formula obtained here, the large N expansion given in [12] does not contain oscillatory (microscopic) terms.

The article is organized as follows. In Section 2, we review the exact expressions of the densities in terms of the generalized Hermite and Laguerre polynomials. In Section 3, we derive the first oscillatory corrections to the global densities (5), (6). These approximations are also compared to the exact densities given in Section 2. The asymptotic densities evaluated about the soft edges of the spectrum are obtained in Section 4; they are expressed in terms of Kontsevich type integrals. The behavior of the latter when the spectral parameter is large is studied in Section 4. In the last section, we finally summarize the principal results and discuss the generalization of some of our results to general β.

Section snippets

Exact expressions of the density

As previously mentioned, the density in the Hermite and Laguerre ensembles can be written as particular generalized Hermite and Laguerre polynomials [1]. These polynomials are symmetric, so we can write them as a linear combination of monomial symmetric functionsmλ(x1,,xN):=x1λ1xNλN+distinct permutations, where λ=(λ1,,λN) is a partition of weight |λ|=i=1Nλi. It is convenient to introduce another basis of the algebra of symmetric polynomials, namely, the (monic) Jack polynomials J¯λ(α). They

Density in the bulk

In this section, we obtain oscillatory corrections to the global densities (5), (6). This is achieved by deforming the contours of integration C in (7) in such a way that they pass through the saddle points of the function f(u,x). In both the Hermite and the Laguerre cases, the function f(u,x) has two simple saddle points in the complex u-plane, called u+ and u. All oscillatory terms can be seen as combinatorial corrections: the global density is recovered when β/2 variables go through u+

Density at the soft-edge

We have seen that the steepest descent method can be applied to the scaled densities only if 1<x<1 (Hermite case) or 0<x<1 (Laguerre case). Indeed, when the spectral parameter x is outside these intervals, the contours of integration cannot be deformed into the steepest decent ones without transgressing the appropriate ordering of the variables of integration. A change of scaling is mandatory. The appropriate changes of variable at the soft edges have been obtained in [6]: the scaled densities

Asymptotics of the Kontsevich type integral

Here we consider the leading order of Kβ,β(x) when x±. This allows to match the soft edge density with the bulk density expanded about the edge.

Proposition 13

When x is large and positiveKβ,β(x)=Γβ,β(2π)βe2β3x3/2x3β/41/2+O(1x3β/4+1).

Proof

Following the discussion in the proof of Proposition 8, we first change the contours in the Kontsevich like integral:Kβ,β(x)=β!(2πi)βV1dv1Vβdvβi=1βevj3/3xvj1j<kβ(vjvk)4/β, where {Vj} is such that V1 goes from eiθ to eiθ, passing through the point x, and such that V

Concluding remarks

The aim of the article was to determine the large-N asymptotic expansion of the density in the Hermite and Laguerre β-ensembles when β2N.

We have shown that the first correction to the global density is purely oscillatory when β>2 and is of order N2/β. In the Hermite ensemble of N×N random matrices, the density contains N peaks; the greater is β and the higher are the oscillations. The influence of the Dyson parameter on the oscillations is the same in the Laguerre ensemble. However, the

Acknowledgements

The work of P.J.F. has been supported by the Australian Research Council. P.D. is grateful to the Natural Sciences and Engineering Research Council of Canada for a postdoctoral fellowship.

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