Renormalized energy concentration in random matrices

TL;DR

This paper introduces a renormalized energy functional to measure disorder in point configurations, computes its expectation for various random processes, and shows concentration and minimization properties for specific point processes.

Contribution

It defines a new energy measure for point configurations, computes its expectation for key random processes, and proves concentration and minimization results.

Findings

Expectation of the renormalized energy is explicitly computed for several processes.

Variance of the energy vanishes, indicating concentration near the mean.

The beta=2 sine process minimizes the energy among translation-invariant determinantal processes.

Abstract

We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix $\beta$-sine processes on the real line (beta=1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the beta=2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.