Abstract
Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut β-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 β-ensemble. As particular cases of the main result we consider the classical β-Gaussian, β-Wishart and β-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.
Highlights
Many textbooks on random matrices [4, 5, 45] begin with the derivation of Wigners semicircle law for the Gaussian unitary ensemble (GUE)5, which is arguably the best known result in random matrix theory (RMT)
Using a combinatorial technique simplified by the sparseness of their matrix models, they managed to obtain formulae for the averages and covariances of the moments
Since the proof is based on the Dyson Coulomb gas analogy it is independent of the particular matrix realization of the ensemble and as such is simpler and more transparent than methods previously available
Summary
Many textbooks on random matrices [4, 5, 45] begin with the derivation of Wigners semicircle law for the Gaussian unitary ensemble (GUE), which is arguably the best known result in random matrix theory (RMT). The main result of the paper is a universal formula for the generating function of (2); it is universal in the sense that it depends only on the support of the equilibrium density, but not on the potential V (x). In principle a complete answer to the general problem of computing spectral linear statistics in the large N limit could be achieved if all functional derivatives with respect to the external confining potential were known. We will restrict ourselves to one-cut ensembles, i.e. we will always assume that the potential V (x) is convex and with superlogarithmic growth at infinity Under these hypotheses, the density of states ρ (x) (15) exists, is absolutely continuous and supported on a single bounded interval supp ρ = [a, b], (a < b), and is the unique probability measure that.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal of Physics A: Mathematical and Theoretical
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
