Research
Probability and statistical mechanics

1. Random geometry and critical phenomena in statistical mechanics

My research is primarily motivated by the study of critical phenomena in statistical mechanics, namely situations in which the same system, governed by the same microscopic laws, organizes itself into qualitatively different macroscopic states. Understanding—through probabilistic and geometric methods—how such dramatic changes, including the emergence of long-range correlations or phase transitions, arise from local interactions is a central and fascinating problem in the field.

To address these questions in a rigorous way, I focus on a range of classical models of statistical mechanics that exhibit critical behavior.

A first class of models consists of spin systems, such as the Ising model and its generalizations to spin O(N) models, which provide fundamental mathematical descriptions of ferromagnetism and collective alignment phenomena. While the Ising model is by now well understood in many regimes, its O(N) generalizations remain significantly less explored from a rigorous mathematical perspective.

I also work on dimer and double-dimer models, which describe systems of diatomic molecules and play a central role in graph theory, and combinatorics.

Another important family of models includes self-avoiding walks and polygons, which serve as basic models for polymers and interacting random walks.

Finally, I study models related to the Bose gas and spatial permutations, which are closely connected to Bose–Einstein condensation. Despite its fundamental importance and the awarding of the Nobel Prize for its physical discovery, a complete and rigorous mathematical understanding of Bose–Einstein condensation in interacting bosons is still missing.

A unifying methodological theme of my work is the use of representations in terms of interacting random walks, loops, or trajectories on a lattice, which provide powerful tools to study correlation decay, phase transitions, and critical behavior.

2. Interacting particle systems and random walks out of equilibrium

I also study stochastic processes and non-equilibrium dynamics arising from interacting particle systems driven by random walks. In contrast to the equilibrium models discussed above, the focus here is on the time evolution of the system, where simple local dynamical rules can produce complex collective behavior over large spatial and temporal scales.

This line of research is motivated by several fundamental questions in probability and statistical physics. One important motivation is to develop a rigorous mathematical understanding of self-organized criticality, a mechanism through which dynamical systems naturally evolve toward critical states without external parameter tuning, and which is believed to play a role in phenomena such as earthquakes, snow avalanches, and other scale-invariant dynamics.. Examples include activated random walks and stochastic sandpiles

Other motivations come from growth and spreading phenomena, such as aggregation processes (IDLA) or branching random walks, which provide simplified mathematical models for epidemic spread and population dynamics.

From a broader perspective, this research also aims at the development of probabilistic techniques—such as coupling methods, random walk estimates, and Green’s function estimates—that can be applied to a wide range of dynamical stochastic models in interacting particle systems.