Characteristic polynomials of random matrices

TL;DR

This paper investigates the universal behavior of characteristic polynomial moments of large random matrices, revealing connections to number theory and providing explicit calculations of these moments in the Dyson scaling limit.

Contribution

It computes the moments of characteristic polynomials as limits of multi-point correlators, demonstrating their universality in the Dyson scaling limit.

Findings

Moments scale as eigenvalue density raised to the power K^2

Universal prefactors independent of specific distributions

Explicit limits of multi-point correlators calculated

Abstract

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices, which are the expectation values of the characteristic polynomials of the matrix. It turns out that these expectation values are quite interesting. For instance, the moments of order 2K scale, for unitary invariant ensembles, as the density of eigenvalues raised to the power $K^2$ ; the prefactor turns out to be a universal number, i.e. it is independent of the specific probability distribution. An equivalent behaviour and prefactor had been found, as a conjecture, within number theory. The moments of the characteristic determinants of random matrices are computed here as limits, at coinciding points, of multi-point correlators of determinants. These correlators are in fact universal in Dyson's scaling limit in which the difference between the points goes to zero, the size of the matrix goes to infinity, and their product remains finite.