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

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 of covariances to those of 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.