Carissimi, Vi ricordiamo che la settimana prossima si terrà la ottava giornata di seminari:

A ``LATE-SUMMER'' DAY IN PROBABILITY AND STATISTICAL PHYSICS

University of Florence Friday 27 September 2019

Lecturers: Pierre Picco (Marseille) and Rui Pires da Silva Castro (Eindhoven)

Location: Aula Tricerri Viale Morgagni 67, Firenze Informazioni su come arrivare alla pagina: https://www.dimai.unifi.it/vp-285-come-arrivare-how-to-get.html https://www.dimai.unifi.it/vp-285-come-arrivare-how-to-get.html

Note pratiche: Stiamo prenotando un catering con cibi vegerariani e non, percio` abbiamo bisogno del numero di persone che vogliono mangiare insieme. A coloro che fossero interessati (per una migliore organizzazione) chiediamo di mandare un email a francescaromana.nardi@unifi.it mailto:francescaromana.nardi@unifi.it; gianmarco.bet@unifi.it mailto:gianmarco.bet@unifi.it; angela.caporicci@unifi.it mailto:angela.caporicci@unifi.it con l'intenzione di partecipare al seminario, al coffe break e al pranzo.

PROGRAMMA Prof. Pierre Picco (CNRS Marseille)

Title: One-dimentional Ising model with long range interactions. A review of results

Abstract: In the first talk I will make a quick historical survey of the rigorous results on the one-dimensional Ising model with long-range interactions.

A first part will be dedicated to uniqueness of the Gibbs states (Ruelle (1968); Dobrushin (1968); Bricmont, Lebowitz & Pfister (1986)) and the regularity of the free energy when the decay of the potential is fast enough (Dobrushin (1973) Cassandro & Olivieri (1981) and its extensions in particular Capocaccia, Campanino & Olivieri (1983).

A second part will be dedicated to the existence of phase transition starting from the Kac-Thompson conjecture (1968) the Dyson results (1969), the Frohlich & Spencer result (1982), the Imbrie result (1982) the Aizenmann, Chayes, Chayes & Newman result on the Thouless effect (1988), Imbrie & Newman result on the Berezinsky, Kosterlitz & Thouless transition (1988).

A third part will be dedicated to present results in the phase transition regime that started with Frohlich & Spencer (1981), Cassandro, Ferrari, Merola & Presutti (2001) and its extensions in particular by Cassandro, Merola, Picco & Rosikov (2014) on the definition of an interface and its fluctuations, and on a Minlos & Sinai theorem on the phase separation problem by Cassandro, Merola & Picco (2017).

In the second talk I will review heuristic arguments that were invoked to conjecture the existence of a phase transition at low temperature in particular the Landau argument. I will present toy models where the fluctuation of interfaces and localisation of the droplet in the Minlos & Sinai theory will be explained. I will give an algorithmic definition of one-dimensional contours of Cassandro, Ferrari, Merola & Presutti.

Prof. Rui Pires da Silva Castro (Eindhoven University of Technology)

Title: Testing for the presence of communities in inhomogeneous random graphs

Abstract: Many complex systems can be viewed as a network/graph consisting of vertices (e.g., individuals) connected by edges (e.g., a friendship relation). Often one believes there is some sort of community structure, where some vertices are naturally grouped together (e.g., more densely connected between themselves than to the rest of the network). Much of the community detection literature is concentrated around methods that extract communities from a given network. Our goal in this work is different, and we attempt to understand how difficult is it to determine if a network has real communities. Furthermore, we are primarily interested in the case of small or very small communities, for which many existing results and methods are not applicable.

We cast this problem as a binary hypothesis test, where the null model corresponds to a graph without community structure, and the alternative model almost the same, but it also includes a planted community - that is, a small subset of the vertices has higher connection probability than under the null. The main question is to determine the minimal size and “strength” of the planted community that will allow detection. The seminal work of Arias-Castro and Verzelen tackled this problem when the null model is a homogeneous random graph. In our work, however, we consider the case where the null model is inhomogeneous, as this is somewhat closer to realistic scenarios. In particular, we present a scan test and provide conditions under which it is able to detect the presence of a small community. These results are valid for a wide variety of parameter choices. Furthermore, we show that for some parameters choices the scan test is optimal, and no other test can perform better (e.g, detect smaller or weaker planted communities). Finally, we extend this scan test to adapt to many parameters of the model when the null is a rank-1 generalized random graph.

In the first part of the talk I will describe the above formulation and ensuing results, with illustrative examples and briefly touching upon the analytical methodology. In addition, I will discuss the related problem of characterizing cliques in rank-1 random graphs, which provides some insights on the role of inhomogeneity. The second part of the talk will go deeper into more technical aspects and ensuing insights. This presentation is based on joint work with Kay Bogerd and Remco van der Hofstad (https://arxiv.org/abs/1805.01688 https://arxiv.org/abs/1805.01688 and ongoing work).

Program: 11.00-11.45 Introductory lecture: Picco 11.45-12.00 Break 12.00-12.45 Seminar: Picco 13.00-14.30 Lunch 14.30-15.15 Introductory lecture: Pires da Silva Castro 15.15-15.30 Break 15.30-16.30 Seminar: Pires da Silva Castro

Organizers: G. Bet, 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.

Ricordo che ciascun 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).

Maggiori informazioni e aggiornamenti sono reperibili alle pagine web

http://web.math.unifi.it/users/fnardi/seminari/ http://web.math.unifi.it/users/fnardi/seminari/ o http://silicio.math.unifi.it/wordpress/probability/days-in-probability-and-s... http://silicio.math.unifi.it/wordpress/probability/days-in-probability-and-statistical-physics/

Vi aspettiamo numerosi Gianmarco Bet e Francesca R. Nardi

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