On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice

Abstract

In this manuscript we study tridiagonal random matrix models related to the classical β-ensembles (Gaussian, Laguerre, Jacobi) in the high temperature regime, i.e. when the size N of the matrix tends to infinity with the constraint that βN " 2α constant, α ą 0. We call these ensembles the Gaussian, Laguerre and Jacobi α-ensembles and we prove the convergence of their empirical spectral distributions to their mean densities of states and we compute them explicitly. As an application we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.

The classical $β$-ensembles with $β$ proportional to $1/N$: from loop equations to Dyson's disordered chain

“…Starting with the work [13], it has been known how to construct random tridiagonal matrices whose eigenvalue probability density function realises the classical β ensembles and thus have functional form given by (2.1) for appropriate w(x). A systematic discussion in the context of the high temperature regime as specified by the relation (1.8) is given in [35]. Our interest for subsequent application is a particular tridiagonal anti-symmetric matrix that gives rise to a variant of (2.1) involving the Laguerre weight, but with squared variables.…”

“…Related to the β-ensembles with the scaling (1.8) are certain classes of random tridiagonal matrices, with i.i.d. entries along the diagonal, and (separately) along the leading diagonal, now referred to as specifying α-ensembles [35].…”

“…After making the N , β-independent change of scale λ j = x j / √ 2 in (1.1), upon the limit (1.8) the the density ρ (1),0 (x) = ρ (1),0 (x; α) is specified by the functional form [2,12,35]…”

“…In the recent work [35] the classical β-ensembles in the high temperature limit have been used to construct a family of tridiagonal matrices referred to as α-ensembles. Moreover, an application was given to the study of generalised Gibbs ensembles associated with the classical Toda lattice [44].…”

High-low temperature dualities for the classical $β$-ensembles

“…It is also the case that the weight function for the orthogonality relation of the associated classical orthogonal polynomials have appeared in studies of the respective β-ensembles in the scaled high temperature regime, β = 2α/N and N → ∞ [3,4,29,38]. Specifically, with the weight function e −βλ 2 /2 modified to e −λ 2 /2 , and with the corresponding PDF then denoted by use of the label "G * " and similarly for the density ρ (1) (x) normalised to integrate to unity, we have that the limit formula [3] ρ G * (1),0 (x; α) := lim…”