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.

Abstract: We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data $\mathcal{H}(0,X)=B(X)$, for $B(X)$ a two-sided standard Brownian motion) and show that as time $T$ goes to infinity, the fluctuations of the height function $\mathcal{H}(T,X)$ grow like $T^{1/3}$ and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift $β$ to the left of the origin and $b$ to the right of the origin. The Fredholm determinant has a limit for $β>b$, and the case where $β=b$ (corresponding to the stationary initial data) follows from an analytic continuation argument.

Abstract: We consider a totally asymmetric simple exclusion on Z with the step initial condition, under the additional restriction that the first particle cannot cross a deterministally moving wall. We prove that such a wall may induce asymptotic fluctuation distributions of particle positions of the form P(supτ∈R{Airy2(τ)−g(τ)}≤S) with arbitrary barrier functions g. This is the same class of distributions that arises as one-point asymptotic fluctuations of TASEPs with arbitrary initial conditions. Examples include Tracy-Widom GOE and GUE distributions, as well as a crossover between them, all arising from various particles behind a linearly moving wall. We also prove that if the right-most particle is second class, and a linearly moving wall is shock-inducing, then the asymptotic distribution of the position of the second class particle is a mixture of the uniform distribution on a segment and the atomic measure at its right end.