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.

arXiv comment33 pages

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Cited by in corpus (14)

• Moments of random matrices and hypergeometric orthogonal polynomials
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Beta Jacobi ensembles and associated Jacobi polynomials
• Differential identities for the structure function of some random matrix ensembles
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Relations between moments for the Jacobi and Cauchy random matrix ensembles
• Moments of discrete orthogonal polynomial ensembles
• Symmetric Function Theory and Unitary Invariant Ensembles
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion
• $\beta$-ensembles and higher genera Catalan numbers