Anisotropic growth of random surfaces in 2+1 dimensions

TL;DR

This paper introduces a family of stochastic growth models in 2+1 dimensions within the anisotropic KPZ class, analyzing their correlation structures, limit shapes, and fluctuation behaviors, revealing connections to the Gaussian free field.

Contribution

It constructs new anisotropic 2+1D growth models with determinantal correlation functions and studies their asymptotic fluctuation properties and limit shapes.

Findings

Surface exhibits a facet-curved shape in the limit.

Height fluctuations are asymptotically normal with ln(t) variance.

Multi-point fluctuations relate to the Gaussian free field on the upper half-plane.

Abstract

We construct a family of stochastic growth models in 2+1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1+1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one-point fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t>>1. (3) There is a map of the (2+1)-dimensional space-time to the upper half-plane H such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H.