A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices

Abstract

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $\beta$-ensemble. As particular cases of the main result we consider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.

16 pages, 4 figures, 3 tables. Revised version where references have been added and typos corrected

References in corpus (3)

• Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices
• Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities
• Joint statistics of quantum transport in chaotic cavities

Cited by in corpus (9)

• Moments of random matrices and hypergeometric orthogonal polynomials
• Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities
• 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
• Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
• Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
• Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices