Disordered systems and (subcritical) polynomial chaos with heavy-tail disorder
Gaspard Gomez
TL;DR
This work develops a general theory for disordered lattice systems with heavy-tailed environments, where the disorder has a finite mean but infinite variance and is attracted to a $\gamma$-stable Lévy white noise with $\gamma\in(1,2)$. By formulating a subcriticality criterion $\lambda<d_{\mathrm{eff}}/\gamma$ and introducing a continuum polynomial chaos expansion, the authors prove the existence and universality of non-trivial scaling limits for the partition function and Gibbs measures, extending prior Gaussian ($L^2$) results to the heavy-tail regime. They rigorously derive continuum disordered objects ${\mathcal Z}_{\Omega}^{\boldsymbol\zeta,\hat{\beta}}$ and continuum disordered measures for two canonical models—the disordered pinning model and the long-range directed polymer—showing convergence under appropriate intermediate-disorder scalings and providing a heavy-tail Harris criterion for disorder relevance. The work also develops robust $L^p$ estimates for chaos with Lévy noise, defines the Lévy chaos, and establishes scaling limits linking discrete polynomial chaos to its continuum counterpart, with universality in the limiting objects. These results connect disordered statistical mechanics to stochastic PDEs in a heavy-tailed setting and broaden the understanding of disorder relevance beyond finite-variance environments. Overall, the paper provides a comprehensive framework for heavy-tail disorder, new continuum limits, and precise criteria governing disorder relevance with potential implications for SPDEs and scaling theory in disordered systems.
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 $P[ω> x] \sim x^{-γ}$ as $x \to +\infty$ for some $γ\in (1,2)$. 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 $γ$-stable Lévy white noise. We develop precise and flexible moments estimates adapted to the heavy-tailed setting.