Disordered systems and (subcritical) polynomial chaos with heavy-tail disorder

TL;DR

This paper investigates disordered statistical mechanics systems influenced by heavy-tailed environments lacking finite second moments, establishing conditions for non-trivial scaling limits and analyzing disorder relevance through polynomial chaos and stable Lévy noise.

Contribution

It introduces a subcriticality condition for heavy-tailed disordered systems and interprets it via a generalized Harris criterion, extending understanding of disorder effects without second moments.

Findings

Established a subcriticality condition for heavy-tailed environments.

Proved the existence of non-trivial scaling limits under certain conditions.

Developed moment estimates for polynomial chaos with heavy-tailed variables.

Abstract

We study discrete statistical mechanics systems perturbed by a random environment without a finite second moment. Specifically, we consider a random environment whose tail distribution satisfies $P[\omega > x] \sim x^{-\gamma}$ as $x \to +\infty$ for some $\gamma \in (1,2)$. Inspired by the seminal work of Caravenna, Sun and Zygouras \cite{csz_2016}, we adopt a general framework that encompasses as key examples both the disordered pinning model and the long-range directed polymer model. We provide some subcriticality condition under which we prove that the discrete disordered system possesses a non-trivial scaling limit. We also interpret the subcriticality condition in terms of a generalized Harris criterion without second moment, which gives a prediction for disorder relevance depending on the parameters of the system. Our analysis relies on the study of multilinear polynomials of independent heavy-tailed random variables known as polynomial chaos and their continuous analogue, given by multiple integrals with respect to a $\gamma$-stable L\'evy white noise. We develop precise and flexible moments estimates adapted to the heavy-tailed setting.