Statistical Mechanics on Random Structures (09w5055)

Objectives

The goal of the workshop is to bring together researchers whose interest lies at the intersection of disordered statistical mechanics and random graph theory, with a clear emphasis on applications. The multidisciplinary nature of the proposed topics will be attracting research groups with different background and will thus ensure an exchange of ideas with cross-fertilisation. As an example, we mention two directions on which we would like to focus during the workshop.

The first example has its origin in the many fundamental issues that are still open in the theory of spin glasses. Even tough today we have a rigorous proof, in the context of mean-field models, of the solution for the free energy first proposed by G. Parisi, certain relevant properties of this solution (e.g. ultrametricity) still lack a firm mathematical understanding. Moreover, when considering spin-glass models on the lattice with short-range interaction, the mean-field picture that predicts the existence of many different equilibrium states is challenged by the droplet scenario, where only a few relevant equilibrium states are assumed (related via the spin-flip symmetry).

The second example concerns the application of ideas and methods from statistical mechanics to the social sciences. Dichotomic social issues (yes or no matters) are known to manifest sudden changes, very much like phase transitions. An approach that has appeared recently is to use mean-field models (i.e., non-local interactions) to describe peer-to-peer relations, possibly extended to social interaction networks carrying the interesting small-world and scale-free features of random graphs. The final aim of this investigation is to establish conditions for opinion spreading, nucleation, cluster formation, and other observed social phenomena, especially within the enormous migratory fluxes nowadays present in developed countries. The use of statistical mechanics for social purposes will also lead to the necessity of statistical parameter estimation via polls, a newly emerging field with several applications.

The workshop will take advantage of an introductory meeting on the same topic that two of the proposers are organising at EURANDOM, Eindhoven, The Netherlands, in March 2008. This meeting (YEP-V, the fifth in a successful series aiming at Young European Probabilists) has a large training component, with many graduate students and postdocs taking part. This will be a partial target for the Banff workshop as well.

Banff 2009 would be a perfect occasion to capitalise on the former meeting, with the perspective to update on ongoing research.