Research

1. Spin systems, random loops, self-avoiding walks, dimer and double-dimer model, random permutations, the Bose gas.

We consider a class of statistical mechanics models which can be described as systems of interacting random walk trajectories. This class includes self-avoiding walks, classical spin systems (e.g. the Ising, XY or Heisenberg model), the double-dimer model, the Bose gas, systems of interacting self-avoiding polygons, the loop O(N) model, lattice permutations. We aim at making progress in understanding the correlation structure, the phase transition phenomenon, and the size and of the geometric properties of the paths. We employ techniques from probability, combinatorics and statistical mechanics.

2. Stochastic processes, particle systems, aggregation process, activated random walks, internal DLA, Abelian networks, self-organised criticality

We consider a class of particle systems driven by random walks. An important feature of such systems is the presence of dynamical constraints (e.g. the number of particles is conserved), which introduce long range correlations. The goal of our research is the understanding of the recurrence properties of such systems, of the phase transition phenomenon, and of the aggregation dynamics. We use probabilistic techniques (random walk properties, essential enhancements, coupling arguments), exploiting the so-called Abelian property of such systems.