Anisotropic (2+1)d growth and Gaussian limits of q-Whittaker processes

TL;DR

This paper studies a discrete anisotropic (2+1)-dimensional growth model linked to q-Whittaker functions, proving convergence to the Gaussian free field and stochastic heat equation in certain limits.

Contribution

It introduces a new discrete growth model connected to q-Whittaker functions and establishes its convergence to Gaussian processes and stochastic PDEs in the scaling limit.

Findings

Bulk height function converges to Gaussian free field.

Model's limits satisfy the (2+1)-dimensional stochastic heat equation.

Explicit integral formulas derived from q-Whittaker functions.

Abstract

We consider a discrete model for anisotropic (2+1)-dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space-time limit to the (2+1)-dimensional additive stochastic heat equation (or Edwards-Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE.