Moments of the Gaussian $\beta$ Ensembles and the large-$N$ expansion of the densities

Abstract

The loop equation formalism is used to compute the $1/N$ expansion of the resolvent for the Gaussian $\beta$ ensemble up to and including the term at $O(N^{-6})$. This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at $O(N^{-6})$. The latter contain non-integrable singularities at the endpoints of the support --- we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings $\beta =1$, $2$ and $4$ there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard we give fifth order linear differential equations for the density and resolvent at $\beta =1$ and $4$, which complements the known third order linear differential equations for these quantities at $\beta =2$.

30 pages, final version has been trimmed and additional moments have been computed, which are recorded in the Appendix

References in corpus (1)

• Computing topological invariants with one and two-matrix models

Cited by in corpus (25)

• Spectral Fluctuations in the Sachdev-Ye-Kitaev Model
• Moments of random matrices and hypergeometric orthogonal polynomials
• Correlators for the Wigner-Smith time-delay matrix of chaotic cavities
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles
• Differential identities for the structure function of some random matrix ensembles
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE
• Quantum curves and conformal field theory
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles
• Loop Equation Analysis of the Circular $\beta$ 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
• Moments of discrete orthogonal polynomial ensembles
• The Harer--Zagier recursion for an irregular spectral curve
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Spectral statistics of the uni-modular ensemble
• The probability distribution of spectral moments for the Gaussian beta-ensembles
• A note on pair dependent linear statistics with slowly growing variance
• Anomalous energy correlations and spectral form factor in the nonergodic phase of the $\beta$-ensemble
• Symmetric Function Theory and Unitary Invariant Ensembles
• Comment on "Finite size effects in the averaged eigenvalue density of Wigner random-sign real symmetric matrices" by G.S. Dhesi and M. Ausloos
• Loop equations for generalised eigenvalue models
• $\beta$-ensembles and higher genera Catalan numbers