Mathematical percolation theory. Theory of random fields.
Theory of stochastic processes. Random walks in random environment.
In the last period I have studied percolation models, in
particular bond and site Bernoulli percolation on a lattice and FK
random cluster models. In an initial work in collaboration
with D. Ioffe we have studied the asymptotic behaviour of
connection probabilities for subcritical Bernoulli percolation and
in particular we have proved Ornstein Zernike behaviour that was
conjectures by the physists Ornstein and Zernike for several
systems and was proved in a previous work of mine in collaboration
with J. Chayes and L. Chayes only in the axes directions of the
lattice. In later works in collaboration with D. Ioffe and Y.
Velenik these results were exteended to correlation functions for
models of dependent random fields (Ising models) amd models of
dependent percolation (FK random cluster models). In a paper in collaboration with M. Gianfelicee appeared on Probability
Theory and Related Fields ideas and methods of previous works are
used for the study of asymptotic behaviour of triple connections of
subcritical Bernoulli percolation . Moreover in collaboration with
D. Ioffe and Lauder we studied a problem related to the
asymptotic behaviour of f inite supercritical conneection
probabilities of two-dimensional Bernoulli percolation. All previous
works are based on suitable probabilistic results: multidimensional
renewal theorems, local limit theorems and large deviation
theorems. In collaboration with M. Gianfelice in two papers we proved Ornstein-Zernike behaviour for finite connection functions of Bernoulli and FK percolation when the parameter p is close to 1.
In collaboration with Dimitri Petritis of the University of Rennes we studied the recurrence properties of random walks on a random two-dimensional graph. In particular we proved almost sure transience that contrasts recurrence in the corresponding deterministic case. Moreover we proved the transition from recurrence to transience for a periodic model with stochastic perturbations decaying with a power law.