Research

• Statistical Mechanics
  (Key words: Spin systems, random loops, self-avoiding walks, dimer and double-dimer models, random permutations, Bose gas)

In this research, we explore a class of statistical mechanics models that can be described as systems of interacting random walk trajectories. This category includes self-avoiding walks, classical spin systems (such as the Ising, XY, and Heisenberg models), the double-dimer model, the Bose gas, interacting self-avoiding polygons, the loop O(N) model, and lattice permutations. Our primary objective is to advance the understanding of the correlation structure, phase transitions, and the geometric properties of these paths. Our approach leverages techniques from probability theory, combinatorics, and statistical mechanics.

• Stochastic Processes
  (Key words: Particle systems, aggregation processes, activated random walks, internal DLA, stochastic sandpile model, critical branching random walks, Abelian networks, self-organized criticality)

We investigate a class of particle systems driven by random walks, where key features include dynamical constraints—such as particle number conservation—that induce long-range correlations. Some of these models, like the activated random walk and the stochastic sandpile model, are particularly relevant in the context of self-organized criticality. Our research aims to deepen the understanding of the recurrence properties of these systems, phase transitions, and aggregation dynamics. We employ probabilistic techniques, including random walk properties, essential enhancements, coupling arguments, and Green's function estimates, while exploiting the Abelian property characteristic of these models.