Root Statistics of Random Polynomials with Bounded Mahler Measure

TL;DR

This paper studies the distribution and local statistics of roots of random polynomials with bounded Mahler measure, revealing new kernel limits and correlations near the unit circle and at special points.

Contribution

It introduces new matrix kernels describing root correlations and analyzes their asymptotics for polynomials with bounded Mahler measure.

Findings

Roots tend to the unit circle as degree grows

New limiting kernels describe local root statistics near special points

Expected number of roots in disjoint regions converges to positive values

Abstract

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree $N$ polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yields a Pfaffian point process on the complex plane. When $N$ is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e., scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of $\pm 1$ which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of $\C$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.