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

In the classical beta-ensembles of random matrix theory, setting beta = 2 alpha/N and taking the N -> infinity limit gives a statistical state depending on alpha. Using the loop equations for the classical beta-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading 1/N correction to the density. From earlier literature, the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying, in particular, the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterization of the moments and covariances of monomial linear statistics is through recurrence relations. We also extend recent work, which begins with the beta-ensembles in the high-temperature limit and constructs a family of tridiagonal matrices referred to as alpha-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d. (Independent Identically Distributed) gamma distributed random variables. From this, we can supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain. Published under an exclusive license by AIP Publishing.

Automated summary

This study investigates statistical states and related properties in classical beta-ensembles, including eigenvalue density, moments, and covariances of monomial linear statistics. The limiting eigenvalue density is found to be related to classical functions, with moments and covariances of monomial linear statistics characterized through recurrence relations. Additionally, an extension of recent work on alpha-ensembles and the construction of random tridiagonal matrices is discussed, providing analytic results for the study of type I disordered chains.