Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations

Abstract

The $\beta$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi $\beta$ ensembles, and use these to systematically construct the explicit form of the $1/N$ expansion at low orders. This allows us to give the explicit form of corrections to the global density, and allows various moments to be computed, complementing results available in the literature motivated by problems in quantum transport.

33 pages

References in corpus (4)

• Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Loop Equation Analysis of the Circular $\beta$ Ensembles
• Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles

Cited by in corpus (7)

• Differential identities for the structure function of some random matrix ensembles
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Relations between moments for the Jacobi and Cauchy random matrix ensembles
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• $\beta$-ensembles and higher genera Catalan numbers
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion