Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime

TL;DR

This paper demonstrates that discretized SHE and KPZ equations in high dimensions share similar fluctuation behaviors in the weak disorder regime, extending previous convergence results and providing new probabilistic bounds.

Contribution

It extends prior convergence results by showing fluctuations of discretized SHE and KPZ are close in probability, under more general conditions.

Findings

Fluctuations of discretized SHE and KPZ are close in probability.

Both equations share the same scaling limit in the weak disorder regime.

Established lower tail concentration for partition functions.

Abstract

We study the fluctuations of discretized versions of the stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation in spatial dimensions $d\geq 3$ in the weak disorder regime. The discretization is defined using the directed polymer model. Previous research has identified the scaling limit of both equations under a suboptimal moment condition and, in particular, it was established that both converge in law to the same limit. We extend this result by showing that the fluctuations of both equations are close in probability in the subcritical weak disorder regime, indicating that they share the same scaling limit (the existence of which remains open). Our result applies under a moment condition that is expected to hold throughout the interior of the weak disorder phase, which is currently only known under a technical assumption on the environment. We also prove a lower tail concentration of the partition functions.