Carissimi, stiamo preparando la prossima giornata di seminari del 25 Maggio 2018 (quarto incontro della serie di giornate sulla probabilita` e le sue applicazioni alla Fisica, Biologia e Scienze sociali). Ricordo che ciascuno oratore fara` una lezione introduttiva e divulgativa di 45 minuti pensata proprio per i non esperti, seguita da altri 45 minuti di tipo seminario (vedi programma).

Per maggiori informazioni e aggiornamenti sulle giornate passate e future abbiamo istituito una pagina web che vi invitiamo a visitare periodicamente. http://web.math.unifi.it/users/fnardi/seminari/

A SPRING DAY IN PROBABILITY AND STATISTICAL PHYSICS

University of Florence Friday 25 May 2018

Lecturers: Francis Comets (Paris) and Remco van der Hofstad (Eindhoven)

Location: Aula Magna di Via S. Gallo 10, Firenze

Prof. Francis Comets (University of Paris Diderot)

Title: Cover time, cover process, random interlacements for random walk on the torus, I and II"

Abstract: The cover time is the time needed for the $d$-dimensional simple random walk to visit all points of the torus of size $n$ on the lattice $Z^d$; in the continuum, it is the time for the Wiener sausage of radius 1 to cover the torus of linear size $n$. As a maximum of correlated random variables (here, with logarithmic decay when $d=2$) it has interesting asymptotics as $n \to \infty$, which are related to the way the covering is performed. Things are different in dimension $d \geq 3$ (the walk is transient) from dimension $d=1,2$ (the walk is recurrent). In dimension $d \geq 3$, Sznitman introduced random interlacements to describe the local picture during the covering process. They consist in a Poissonian soup of bi-inifinite random walk trajectories. They still give a reliable account at large densities, up to cover time. We will emphasize dimension 2: random interlacements can be used as a description of the neighborhood of an unvisited site, provided that the paths used in the interlacements are random walk _conditioned not to visit a point_. (Joint works with Serguei Popov and Marina Vachkovskaia.) (Leiden University)

Prof. Remco van der Hofstad (Technical University Eindhoven)

Title: Ising models on random graphs

ABSTRACT: The Ising model is one of the simplest statistical mechanics models that displays a phase transition. While invented by Ising and Lenz to model magnetism, for which the Ising model lives on regular lattices, it is now widely used for other real-world applications as a model for cooperative behavior and consensus between people. As such, it is natural to consider the Ising model on complex networks. Since complex networks are modelled using random graphs, this leads us to study the Ising model on random graphs. In this talk, we discuss some recent results on the stationary distribution of the Ising model on locally tree-like random graphs. We start by giving an extensive introduction to random graph models for complex networks, to set the stage of the graphs on which our Ising models live. Real-world networks tend to be highly inhomogeneous, a fact that is most prominently reflected in their degree distributions having heavy tails as described by power laws. Due to the randomness of the graphs on which the Ising model lives, there are different settings for the Ising model on it. The quenched setting describes the Ising model on the random graph as it is, while the averaged quenched setting takes the expectation w.r.t. the randomness of the graph. As such, it takes the expectation of the Boltzman distribution, which is a ratio of an exponential involving the Hamiltonian, and the partition function. In the annealed setting, the expectation is taken on both sides of the ratio. These different settings each describe different physical realities. We discuss the thermodynamic limit of the Ising model, which can be used to define the phase transition in the Ising model on locally tree-like random graphs, by describing when spontaneous magnetization exists and when not, extending work by Dembo and Montanari. We give an explicit expression for the critical value and the critical exponents for the magnetization close to it. These critical exponents depend on the power-law exponent of the degree distribution in the random graph. We also discuss central limit theorems for the total spin in the uniqueness regime, as well as a non-classical limit theorem for the total spin at the critical point in the special setting of the annealed generalized random graph. This talk is based on several joint works with Sander Dommers, Cristian Giardina, Claudio Giberti and Maria Luisa Prioriello.

Program: 11.00-11.45 Introductory lecture: Comets 11.45-12.00 Break 12.00-13.00 Seminar: Comets 13.00-14.30 Lunch 14.30-15.15 Introductory lecture: van der Hofstad 15.15-15.30 Break 15.30-16.30 Seminar: van der Hofstad

Organizers: F. Caravenna, N. Cancrini, E.N.M. Cirillo, P. Dai Pra, A. De Masi, D. Fanelli, F. Flandoli C. Giardina`, R. Livi, F. Martinelli, I.G. Minelli, F.R. Nardi, E. Presutti, B. Scoppola, E. Scoppola

Note pratiche: 1) E` importante prenotare il biglietto del treno il piu` presto possibile per poter usufruire di sconti!!!! 2) Stiamo prenotando un ristorante (diverso dal precedente) con un paio di menu fissi (di cui uno vegerariano) che possa ospitarci, ma ho bisogno del numero di persone che vogliono mangiare insieme. A coloro che fossero interessati chiedo di mandare un email a francescaromana.nardi@unifi.it Resta fermo il fatto che saremo in pieno centro di Firenze, quindi ci sono moltissime altre possibilita` per mangiare se preferite regolarvi indipendentemente.

Vi aspettiamo numerosi Francesca R. Nardi

Dipartimento di Matematica e Informatica Università degli Studi di Firenze Viale Morgagni 67, Firenze, Italy