Marzio Cassandro

Metastable behavior of stochastic dynamics: A pathwise approach

In this paper a new approach to metastability for stochastic dynamics is proposed. The basic idea is to study the statistics of each path, performing time averages along the evolution. Metastability would be characterized by the fact that the process of these time averages converges, under a suitable rescaling, to a measure valued Markov jump process. Here this convergence is shown for the Curie-Weiss mean field dynamics and also for a model with spatial structure: Harris contact process.

Surface tension in Ising systems with Kac potentials

We consider an Ising spin system with Kac potentials in a torus of ℤd,d>-2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter γ vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they Γ-converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our considerations include nonsmooth interfaces, proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.

Some probabilistic techniques in field theory

We study, in the context of the Markov hyerarchical fields (d=2, 3) the role of the Markov property, of formal renormalization and of formal positivity. We determine upper and lower bounds for the ground state energy and discuss their relation with the perturbation theory series.

Existence and uniqueness of DLR measures for unbounded spin systems

SummaryA system of random variables(spins) Sx,x∈ℤ v, taking on values in ℝ is considered. Conditional probabilities for the joint distributions of a finite number of spins are prescribed; a DLR measure is then a process on the random variables which is consistent with the assigned conditional probabilities [1,2]. A case of physical interest both in Statistical Mechanics and in the lattice approximation to Quantum Field Theory is considered for which the spins interact pairwise via a potential JxySxSy, Jxy∈ℝ and via a self-interaction F(Sx), which, as ¦Sx¦→∞, diverges at least quadratically [3].By use of a technique introduced in [2] it is proven that the set

A study of metastability in the Ising model

\mathfrak{E} = \{ v is DLR|\exists c(v), \mathop {sup}\limits_{x \in \mathbb{Z}^v } \int {v(dS_x )} |S_x | < c(v)\}

On the existence of thermodynamics for the generalized random energy model

is a compact (in the local weak topology, Def. 1.1) non-void Choquet simplex [4]. Sufficient conditions are then given in order to obtain the measures in

Typical Configurations for One-Dimensional Random Field Kac Model

\mathfrak{E}