Large deviations for zeros of $P(\phi)_2$ random polynomials

TL;DR

This paper extends large deviations principles for zeros of Gaussian random polynomials to non-Gaussian ensembles inspired by quantum field theory, showing similar asymptotic behavior and equilibrium measures.

Contribution

It introduces $P()_2$ random polynomials, a non-Gaussian ensemble, and proves large deviations principles with the same rate function as the Gaussian case.

Findings

Large deviations principles hold for $P()_2$ ensembles.

Expected zero distribution converges to the same equilibrium measure as Gaussian case.

Speed and rate function match those of Gaussian ensembles.

Abstract

We extend results of Zeitouni-Zelditch on large deviations principles for zeros of Gaussian random polynomials $s$ in one complex variable to certain non-Gaussian ensembles that we call $P(\phi)_2$ random polynomials. The probability measures are of the form $e^{- S(f)} df$ where the actions $S(f)$ are finite dimensional analgoues of those of $P(\phi)_2$ quantum field theory. The speed and rate function are the same as in the associated Gaussian case. As a corollary, we prove that the expected distribution of zeros in the $P(\phi)_2$ ensembles tends to the same equilibrium measure as in the Gaussian case.