Gaussian Limits for Subcritical Chaos

TL;DR

This paper introduces a second-moment based criterion for Gaussian convergence in chaos, applying it to directed polymers, KPZ, and stochastic heat equations, and provides a new chaos expansion for the partition function's logarithm.

Contribution

It offers a novel, simple criterion for Gaussian limits in chaos and derives a new explicit chaos expansion for the logarithm of the partition function.

Findings

Gaussian limits for subcritical chaos established

New asymptotics for directed polymer partition functions

Explicit chaos expansion for the log-partition function

Abstract

We present a simple criterion, only based on second moment assumptions, for the convergence of polynomial or Wiener chaos to a Gaussian limit. We exploit this criterion to obtain new Gaussian asymptotics for the partition functions of two-dimensional directed polymers in the sub-critical regime, including a singular product between the partition function and the disorder. These results can also be applied to the KPZ and Stochastic Heat Equation. As a tool of independent interest, we derive an explicit chaos expansion which sharply approximates the logarithm of the partition function.