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Research interests

My research interests lie mainly in the applications of Probability Theory to the study of rigorous classical Statistical Mechanics, especially lattice random fields (spin systems, effective interface models, etc.), and random walks (self-interacting random walks, polymers, etc.). One recurrent theme in my research is the derivation of the large-scale asymptotics of various extended objects, such as interfaces, polymers, etc., and the associated phase transitions. Below is a short summary of some of the results obtained (a more detailed account of papers [1] to [18] can be found in the research summary I wrote for my habilitation, see here; you may also have a look at these lecture notes).

Ornstein-Zernike asymptotics


A celebrated heuristic theory proposed by Ornstein and Zernike in 1914 implies that the asymptotic form of the truncated two-point density correlation function of simple fluids away from the critical region is given by \[ G_\beta(r) = A_\beta r^{-(d-1)/2} e^{-\xi_\beta r}, \] where the value of the inverse correlation length $\xi_\beta$ depends only on the inverse temperature and the spatial dimension $d$. In several papers in collaboration with M. Campanino and D. Ioffe, we developed a non-perturbative approach to derive rigorously such estimates in various systems. An introduction can be found here. Among the results obtained are:

Effective interface models


Up to now, the analysis of the interactions of rough interfaces with various external potentials (pinning, wetting, etc.) is too hard to be carried through rigorously for lattice spin systems. For this reason, it is useful to consider simplified models of interfaces, the so-called effective interface models. In these models, interfaces are represented as graphs of (random) functions from an underlying lattice to $\mathbb{R}$ or $\mathbb{Z}$. I have been interested, with various collaborators, in the behavior of such systems. Among the results obtained are:

Polymers


The probabilistic analysis of simple effective models for polymers has been a very active field of study in recent years. The topics I have worked on, with various collaborators, include:

Phase separation in lattice spin systems


These papers are about the properties of interfaces in lattice spin systems.

Others


The following are various other topics I have worked on, with various collaborators.