The reciprocal Mahler ensembles of random polynomials

TL;DR

This paper studies the roots of reciprocal random polynomials with bounded Mahler measure, revealing their convergence to an arcsine distribution and introducing new kernels related to Ginibre ensembles, with detailed analysis of their local behavior.

Contribution

It generalizes Ginibre ensembles to reciprocal polynomials, derives new kernels, and analyzes their asymptotic behavior and outliers, expanding understanding of root distributions in random polynomial ensembles.

Findings

Roots form determinantal and Pfaffian point processes

Empirical measure converges to the arcsine distribution on [-2,2]

New kernels are introduced and compared with classical kernels

Abstract

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree $N$ whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre's complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval $[-2,2]$ on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on $[-2,2]$. Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from $[-2,2]$. These kernels, as well as the scaling limits for the kernels in the bulk $(-2,2)$ and at the endpoints $\{-2,2\}$ are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.