**ABSTRACT**

In this talk I will present some limit theorems for the total spin of ferromagnetic Ising models on random graphs. In such systems two different source of randomness are present: the Ising spins that are random variables distributed according to the Boltzmann-Gibbs measure and the spatial disorder given by the random graph. This is the source of distinction between the "random quenched measure", the "average quenched measure" and the "annealed measure". In the uniqueness regime, the Central Limit Theorem (CLT) can be proven for the general tree-like graphs in the random quenched setting. The average quenched CLT is much more challenging, thus we restrict ourselves to two "configuration models": the model with all vertices having degree 2 (CM(2)) and the case where vertices have either degree 1 or degree 2 (CM(1,2)). In the annealed setting the CLT can be proven for the Ising model on the CM(2), CM(1,2) graphs and also on the Generalized Random Graph (GRG). The Ising model on the GRG graph has a finite critical temperature: I will discuss a non-classical limit theorem at the critical point, where the CLT breaks down, and I will show that the critical exponents match those of an Ising model on locally tree-like random graph in the random quenched setting. (joint work with S. Dommers, C. Giardinà, R. van der Hofstad, M.L. Prioriello)

**OTHER SPEAKERS**

- Cristian Giardinà (University of Modena)
- Cecilia Vernia (University of Modena)

**RESEARCH PROGRAM**

This seminar is part of a research program on “Self sustained currents”. In a recent paper, “Latent heat and the Fourier law” by M. Colangeli, A. De Masi and E. Presutti, (to appear on Phys. Letters A), it has been defined a cellular automaton which describes the evolution of particles in a circuit which have a non zero current despite the fact that no external force acts on the particles. Our program is to study the model numerically and theoretically and to investigate it for a larger set of values of the parameters; in perspective we want to study two dimensional systems. Listed below are the researchers involved in the program:

- Matteo Colangeli (GSSI)
- Anna De Masi (University of L’Aquila)
- Cristian Giardinà (University of Modena)
- Claudio Giberti (University of Modena)
- Errico Presutti (GSSI)
- Cecilia Vernia (University of Modena)