Correlations between zeros and supersymmetry

TL;DR

This paper extends the computation of universal correlation limits for zeros of random polynomials across all dimensions and codimensions, employing supersymmetry and Berezin integrals to derive explicit formulas.

Contribution

It introduces a supersymmetry-based method to compute universal correlation limits for zeros of random polynomials in all dimensions and codimensions, expanding previous results.

Findings

Explicit formulas for pair correlations in all dimensions and codimensions.

Use of supersymmetry and Berezin integrals to derive correlation functions.

Universal scaling limits for zeros of random polynomials.

Abstract

In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions.