Immacolata Merola

Geometry of contours and Peierls estimates in d=1 Ising models with long range interactions

Following Frohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as ∣x−y∣−2+α, 0⩽α⩽1∕2. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well-known result by Dyson about phase transitions at low temperatures.

Coexistence of Ordered and Disordered Phases in Potts Models in the Continuum

This is the second of two papers on a continuum version of the Potts model, where particles are points in ℝd, d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ−1, γ>0. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist S+1 mutually distinct DLR measures.

Potts Models in the Continuum. Uniqueness and Exponential Decay in the Restricted Ensembles

In this paper we study a continuum version of the Potts model, where particles are points in ℝd, d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ−1, γ>0. In mean field, for any inverse temperature β there is a value of the chemical potential λβ at which S+1 distinct phases coexist. We introduce a restricted ensemble for each mean field pure phase which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin-Shlosman theory (Dobrushin and Shlosman in J. Stat. Phys. 46(5–6):983–1014, 1987), we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for γ small enough which implies uniqueness and exponential decay of correlations. In a second paper (De Masi et al. in Coexistence of ordered and disordered phases in Potts models in the continuum, 2008), we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.

First-Order Phase Transition in Potts Models with Finite-Range Interactions

We consider the Q-state Potts model on Zd, Q≥ 3, d≥ 2, with Kac ferromagnetic interactions and scaling parameter γ. We prove the existence of a first order phase transition for large but finite potential ranges. More precisely we prove that for γ small enough there is a value of the temperature at which coexist Q+1 Gibbs states. The proof is obtained by a perturbation around mean-field using Pirogov-Sinai theory. The result is valid in particular for d = 2, Q = 3, in contrast with the case of nearest-neighbor interactions for which available results indicate a second order phase transition. Putting both results together provides an example of a system which undergoes a transition from second to first order phase transition by changing only the finite range of the interaction.

Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas

In this paper we study the statistics of combinatorial partitions of the integers, which arise when studying the occupation numbers of loops in the mean field Bose gas. We review the results of Lewis and collaborators and get some more precise estimates on the behavior at the critical point (fluctuations of the condensate component, finite volume corrections to the pressure). We then prove limit shape theorems for the loops occupation numbers. In particular we prove that in a certain range of the parameters, a finite fraction of the total mass is, in the limit, supported by infinitely long loops. We also show that this mass is equal to the mass of the condensed state where all particles have zero momentum.

Phase Transitions in Layered Systems

We consider the Ising model on

Layered Systems at the Mean Field Critical Temperature

\mathbb Z\times \mathbb Z

One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point

Z×Z where on each horizontal line