MARK KAC SEMINAR

There is a somewhat unexpected connection between random walks in time-dependent random environments and gradient Gibbs measures describing stochastic interfaces in systems arising from statistical mechanics, e.g., the Ginzburg-Landau model, and its dynamics. After reviewing how the space-time covariances of the height of the interface can be expressed in terms of random walks among dynamic random conductances, I will discuss recent progress on the understanding of the behaviour of such random walks. A particular emphasis will be on the results and the methods that has been used to prove invariance principles, local limit theorems and heat estimates for almost every realisation of the environment.

Random interfaces arise naturally as separating surfaces between two different thermodynamic phases or states of matter, for example oil and water. In this talk we will introduce a few Gaussian models for random interfaces: they are called the discrete Gaussian free field (DGFF), the membrane model (MM), and the $\nabla+\Delta$-model, which represents a mixture between DGFF and MM. We will present their similarities and differences, and in addition we will discuss their connections to the theory of partial differential equations and numerical analysis. Later we will focus on their scaling limits, and we will show in particular that in the mixed case the GFF component of the model dominates in the limit. Based on joint works with Biltu Dan and Rajat Subhra Hazra (ISI Kolkata).