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

TL;DR

The paper advances the understanding of high-dimensional stochastic growth by showing that discretizations of the stochastic heat equation (SHE) and the KPZ equation, implemented via directed polymers in $d\ge 3$, have fluctuations that are close in probability in the subcritical weak-disorder regime. By developing a robust $L^p$-based approximation framework and leveraging Burkholder-type inequalities, the authors connect the nonlinear KPZ discretization to the linear SHE-like fluctuations, establishing a precise comparison between $S_n(f)$ and $K_n(f)$ and proving a lower-tail concentration for the partition function. The main result, under a subcritical moment assumption and a mild environmental concentration property, states that $|S_n(f)-K_n(f)|$ decays faster than $n^{-\xi}$ up to a small loss $n^{-\varepsilon}$, implying the two discretizations share the same scaling behavior and potentially the same scaling limit. This provides evidence that the Edwards–Wilkinson-type universality in higher dimensions can manifest for discretized KPZ models and paves the way for identifying the limiting (possibly Lévy) fluctuations in the weak-disorder regime, with substantial implications for numerical discretizations and theoretical analyses of high-dimensional SPDEs.

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.

Theorems & Definitions (47)

• Remark 1.1
• Theorem A
• Theorem B: junk2023local
• Theorem 1.2
• Theorem C
• Theorem D
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Corollary 1.6