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 $\beta$-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 $\beta N=2\alpha$ constant, $\alpha >0$. We call these ensembles the Gaussian, Laguerre and Jacobi $\alpha$-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.

21 pages, 3 figures. We add some new references

References in corpus (4)

• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Beta Jacobi ensembles and associated Jacobi polynomials
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain

Cited by in corpus (8)

• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Equilibrium Spacetime Correlations of the Toda Lattice on the Hydrodynamic Scale
• Large Deviations for Ablowitz-Ladik lattice, and the Schur flow
• Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, Circular $\beta$-ensemble and double confluent Heun equation
• Dyson's disordered linear chain from a random matrix theory viewpoint
• Discrete integrable systems and random Lax matrices
• CLT for real beta-ensembles at high temperature
• Generalized Hydrodynamics for the Volterra lattice: Ballistic and nonballistic behavior of correlation functions