MARK KAC SEMINAR

In the first part of this presentation, we review some recent progress in deriving power series representations for various t hermodynamic quantities such as pressure, free energy and correlation functions for inhomogeneous systems of i nteracting particles in a continuous medium. We use cluster expansion techniques in the context of generating functions of various combinatorial species. In the second part we consider a binary system of repelling small and large (hard) spheres in a continuous medium. This could describe colloidal particles (large spheres) within a substrate (small spheres). One interesting phenomenon is that despite the originally repulsive forces between all particles, when we look at the effective system of only large spheres, an attractive force emerges between them, usually referred to as "depletion attraction". We will discuss a sufficient condition for the convergence of the related cluster expansion that involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods to the binary system). This is based on joint works with Sabine Jansen, Giuseppe Scola and Xuan Nguyen.

Let G be a transitive graph of polynomial growth (the lattice Z^d in the simplest case, or the Cayley graph of H_3(Z^d) the discrete Heisenberg group in the simplest non-abelian case). By the theorems of Gromov, Trofimov, Bass, and Guivarc'h, the graph G has a well-defined dimension d . Long-range percolation on G is the random graph where we include an edge between two distinct vertices x,y \in G with probability 1 - \exp(-\beta d_G(x,y)^{-d \alpha}) where \beta \geq 0 is the edge-density parameter, \alpha > 1 is the long-range parameter, and d_G is the graph metric. Let \mathbf{P}_{\beta} be the law of this random graph and let K be the cluster of the origin. It follows from classical results that this model undergoes a phase transition at \beta_c = \inf\{\beta \geq 0 : \mathbf{P}_{\beta}(\# K = \infty) > 0 \} which satisfies 0 < \beta_c < \infty if d \geq 2 and \alpha > 1 or d = 1 and 1 < \alpha \leq 2. In this talk we will be interested in the supercritical regime \beta > \beta_c when there exists a unique infinite cluster. The starting point will be the truncated one-arm event \{o \leftrightarrow B(r)^c, \# K < \infty\} that there exists an open path from the origin to the complement of the ball B(r) and the cluster of the origin K is finite. Whereas in Bernoulli nearest-neighbour percolation \{o \leftrightarrow B(r)^c, \# K < \infty\} implies \{r \leq \# K < \infty \}, in long-range percolation edge lengths are unbounded, and it is natural to ask what is more likely: a single long edge? Several shorter edges? Or a combination of both? On the way to answering this question, we will discuss several quantitative results in the supercritical regime, focusing in particular in the setting 1 < \alpha < 2 where we are able to use powerful renormalisation techniques. As a consequence of these quantitative results, we prove an analogue of Schramm's locality conjecture, which says in a certain precise sense that the value of the critical parameter \beta_c is determined by the local structure of the graph. This talk is based on joint work with J\'ulia Komj\'athy.

We introduce the phenomenon of metastability through examples from spin systems and methods from potential theory. We then review the main techniques used to establish it. In the second part, we present the two-pattern Hopfield model and show that it can be viewed as two independent Curie–Weiss models at possibly different effective temperatures. Motivated by this analogy, we outline a proof of metastability for systems composed of two independent subsystems.