Stochastic heat equation limit of a (2+1)d growth model

TL;DR

This paper analyzes the limit of a 2D growth model related to the anisotropic KPZ class, showing Gaussian fluctuations and heat equation behavior, confirming the irrelevance of non-linearity in this context.

Contribution

It establishes the $q o 1$ limit of a 2D $q$-Whittaker particle system and connects it to the stochastic heat equation, advancing understanding of anisotropic KPZ universality.

Findings

Gaussian free field fluctuations in stationary measures

Asymptotic correlations match 2D stochastic heat equation

Non-linearity in 2D anisotropic KPZ is irrelevant

Abstract

We determine a $q\to 1$ limit of the two-dimensional $q$-Whittaker driven particle system on the torus studied previously in [Corwin-Toninelli, arXiv:1509.01605]. This has an interpretation as a $(2+1)$-dimensional stochastic interface growth model, that is believed to belong to the so-called anisotropic Kardar-Parisi-Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the $(2+1)$-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in $(2+1)$-dimension is irrelevant.