Conference: Workshop on the occasion of Erwin Bolthausen's 70th birthday


A class of (2+1)-dimensional growth process with explicit stationary measure

{{_Ltalk:R}} Prof. Dr. Fabio Toninelli
Date: 15.09.16   Time: 16.20 - 17.10   Room: Y16G05

We introduce a class of $(2 + 1)$-dimensional random growth processes, that can be seen as asymmetric random dynamics of discrete interfaces. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. “Asymmetric” means that the interface has an average non-zero drift. When the asymmetry parameter $p − 1/2$ equals zero, the infinite-volume Gibbs measures $pi_\rho (with given slope \rho)$ are stationary and reversible. When $p\neq 1/2, \pi_\rho$ is not reversible any more but, remarkably, it is still stationary. In such stationary states, one finds that the height function at a given point $x$ grows linearly with time $t$ with a non-zero speed,

$<(h_x(t)-h_x(0))>= vt$

and that the typical fluctuations of $(h_x(t)-h_x(0))$ are smaller than $\sqrt{\log t}$. For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” studied by A. Borodin and P. L. Ferrari.

If time allows, I will also mention related recent developments on convergence to the stochastic heat equation for a related model (joint work with A. Borodin and I. Corwin).