Foto del docente

Massimo Campanino

Professor

Department of Mathematics

Academic discipline: MAT/06 Probability and Statistics

Research

Keywords: Percolation Random cluster representation Stochastic Processes Random fields Local limit theorems Markov chains Renewal theory

 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.