Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles

Abstract

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical $\beta$-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.

LaTeX, 26 pages, 2 figure. Revised version. Minor corrections

References in corpus (9)

• Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
• Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry
• Jack polynomials and free cumulants
• A shortcut through the Coulomb gas method for spectral linear statistics on random matrices
• Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes
• Correlators for the Wigner-Smith time-delay matrix of chaotic cavities
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices
• Loop Equation Analysis of the Circular $\beta$ Ensembles

Cited by in corpus (10)

• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble
• 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
• Spectral statistics of the uni-modular ensemble
• Symmetric Function Theory and Unitary Invariant Ensembles
• Schur expansion of random-matrix reproducing kernels
• $\beta$-ensembles and higher genera Catalan numbers
• Improved concentration of Laguerre and Jacobi ensembles