MARK KAC SEMINAR

Interacting particle systems may exhibit, in the thermodynamic limit, a time-periodic behavior in the evolution of their law. We have made an attempt to understand which factors may produce this phenomenon, in particular those related to the time-symmetry breaking of the dynamics: dissipation, delay, asymmetry in the interaction. We will first review some examples of mean field dynamics for which the thermodynamic limit can be explicitly computed and analyzed. Later we will focus on some partial results concerning the first, to our knowledge, example of rhythmic behavior for a system with local interaction: the Ising model with dissipation.

We consider a random walk on the d-dimensional lattice in the random conductance model. Each edge of the lattice is assigned randomly a conductance and for a fixed realization of this environment, the random walker crosses an edge with a probability proportional to the conductivity of the edge. This model is one of the prime examples of a reversible process in an inhomogeneous medium. When we introduce a bias to the right, the process satisfies a law of large numbers with a nonzero effective speed. We are interested in properties of the speed as a function of the bias. For example, is the speed continuous, and is it increasing in the strength of the bias?

We will discuss general ideas how to deal with such a random medium and how it can lead to some atypical behavior. The talk is based on joint works with Noam Berger, Nina Gantert and Xiaoqin Guo.