On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window

TL;DR

This paper analyzes the critical behavior of the (2+1)-dimensional directed polymer and stochastic heat equation, computing moments and establishing non-trivial limits in a weak disorder regime.

Contribution

It provides explicit calculations of the third moment and shows the existence of non-trivial subsequential limits for the rescaled partition functions at criticality.

Findings

Third moment of the partition function is uniformly bounded.

Rescaled partition functions have non-trivial subsequential limits.

Limits share an explicit covariance structure.

Abstract

The partition function of the directed polymer model on Z^{2+1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R^2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochastic heat equation on R^2, extending previous work by Bertini and Cancrini.