Polynomial chaos and scaling limits of disordered systems

TL;DR

This paper develops a unified framework using polynomial chaos expansions and a Lindeberg principle to analyze the scaling limits of disordered systems in statistical mechanics, revealing new continuum models.

Contribution

It introduces general conditions for convergence of polynomial chaos expansions to Wiener chaos limits, extending previous work and applying to various disordered models.

Findings

Unified framework for continuum and weak disorder limits

Extension of Lindeberg principle to polynomial chaos

Identification of new continuum models for disordered systems

Abstract

Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.