MARK KAC SEMINAR

In this talk we will consider discrete metric trees (T,r) together with a measure ν and the random walk on (T,r) which jumps from a vertex v∈T to a neighboring vertex v'∼v with rate (ν({v})⋅r(v,v'))^-1.

We construct a potential limiting process on a locally compact R-tree (T,r) equipped with a Radon measure ν which we call ν-speed Brownian motion on (T,r). We will show that a family of ν_n-speed random walks on (T_n,r_n) converges in path space to the ν-speed Brownian motion provided that the underlying metric measure spaces (T_n,r_n,μ_n) converge towards (T,r,ν) in the Gromov–Hausdorff-vague topology. The topology will be introduced in the talk and its relation to the weaker Gromov-vague topology will be discussed. In fact, there is a uniform lower bound property for the family (ν_n) which together with Gromov-vague convergence is equivalent to Gromov–Hausdorff-vague convergence. If we only assume convergence in Gromov-vague topology, our processes might not convergence in path space anymore due to paths possibly entering regions of ν-zero measure but we do still get convergence in f.d.d.

(This is joint work with Siva Athreya and Wolfgang Löhr)

We consider a system of particles in R^d interacting via a reasonable potential with both long and short range contributions and prove rigorously the existence of a liquid-vapor branch in the phase diagram of fluids.

The model we consider is a variant of the model introduced by Lebowitz, Mazel and Presutti (1999), obtained by adding a hard core interaction to the original Kac potential interaction, the first acting on a scale of order 1 with respect to the Kac parameter. We prove perturbatively that if the hard core radius R is sufficiently small, then the liquid-vapor phase transition proved for the LMP model is essentially unaffected. Hence, we prove existence of two different Gibbs measures corresponding to the two phases. This is a joint work with Errico Presutti and Dimitrios Tsagkarogiannis.