51³Ô¹ÏÍø

Title: Annealed Potts models on rank-1 random graphs

Abstract: In this talk, we look at the annealed ferromagnetic $q$-state Potts model, on sparse rank-1 random graphs, $q \geq 3$. After motivating the problem and looking a little at the Ising case for the sake of comparison, we will discuss our main results and time permitting, some proof techniques. Our results include convergence of the pressure per particle for a rather general class of weights, no phase transition for infinite variance weights — the system is always ordered for any positive temperature, for finite-variance weights — a proof of the first order phase transition at the critical temperature under a quite general condition, and for the special case of Pareto weights with power law exponent $\tau$ — a `phase transition smoothening' — the phase transition is first order when $\tau \geq 4$, while when $\tau \in (3,4)$, it is first order when $\tau \in (\tau(q),4), while it is second order for $\tau \in (3, \tau(q)]$. We end with a discussion on future extensions and open problems. This is based on joint work with Cristian Giardinà (Modena), Claudio Giberti (Modena), Remco van der Hofstad (Eindhoven) and Guido Janssen (Eindhoven).