I am associate professor ("maître de conférences") in probability at Ceremade, the mathematics department of Université Paris-Dauphine.
Contact
- E-mail: last_name @ ceremade . dauphine . fr
- Address: Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75 775 Paris Cedex 16, France.
Teaching
Control of Markov chains (M1)
The teaching material is available on Teams.A review of probability theory foundations (M2 MATH)
Lecture notes (please signal any mistakes)
Research
My research work deals with discrete probability models connected to statistical mechanics:
- Activated random walks
- Stochastic sandpile model
- Self-organized criticality
- Interacting random loops
Activated random walks
The activated random walks model is a system of interacting particles in which particles perform random walks on a graph, falling asleep at a certain rate and being waken up when another particle arrives on the same site.
This model emerged as a variant of sandpile models, which were suggested by physicists to illustrate the concept of self-organized criticality.
There is a phase transition: a critical density separates a stable phase (where particles eventually fall asleep) and an active phase (where activity is maintained).
With Alexandre Gaudillière and Amine Asselah, we showed that in dimension 2 the critical density is strictly below 1, concluding a series of works aimed at demonstrating the non-triviality of the phase transition in any dimension.
|
Active Phase for Activated Random Walks on the Lattice in all Dimensions
Ann. Inst. Henri Poincaré Probab. Stat. 60 (2024), no. 2, 1188–1214 |
While the above article deals with the case of small sleep rate, the following publication tackles the more general case:
|
The Critical Density for Activated Random Walks is always less than 1
Ann. Probab. 52(5): 1607-1649 (September 2024) |
|
The following work shows that the stabilization of a segment containing a supercritical density causes a fraction of particles to exit at the endpoints. It can be seen as a first step towards the density conjecture, which connects different definitions of the critical density of the model.
|
Macroscopic flow out of a segment for Activated Random Walks in dimension 1
ALEA, Lat. Am. J. Probab. Math. Stat. 22, 557-577 (2025) |
This density conjecture was proved in dimension 1 by Christopher Hoffman, Tobias Johnson and Matthew Junge. The following article presents a new proof of this result, based on a superadditivity property for the stationary density of the model (while the work of Hoffman, Johnson and Junge relies on superadditivity for a different quantity, which is indirectly connected with the stationary density).
|
A new proof of superadditivity and of the density conjecture for Activated Random Walks on the line Preprint (2025) |
|
With Christopher Hoffman, Tobias Johnson, Josh Meisel, Jacob Richey and Leonardo Rolla, we extended a result of Madeline Brown, Christopher Hoffman and Hyojeong Son about explosivity of the model in dimension 1 : starting from a supercritical configuration, one single active particle is enough to wake up all the other particles with positive probability.
|
Explosivity in 1-d Activated Random Walk
Preprint (2026) |
Stochastic sandpile model
The stochastic sandpile model is a close variant, which can be seen as an intermediate between activated random walks and the abelian sandpile, the model initially suggested by physicists which launched this line of research.
With Concetta Campailla and Lorenzo Taggi, we showed that the critical density of this model is strictly less than 1 in any dimension:
|
The critical density of the Stochastic Sandpile Model
Preprint (2024) |
|
The following article explains how to sample from the stationary distribution of this model on any graph, and studies the case of the complete graph:
|
Stochastic Sandpile model: exact sampling and complete graph
to appear in Electron. J. Probab. (2025) |
Self-organized criticality
The concept of self-organized criticality aims at describing some physical systems which present a “critical” behaviour without having to tune a parameter (like temperature) to a critical value. This is one of the motivations to study the sandpile models and activated random walks.
With Raphaël Cerf (my PhD advisor), we constructed several toy models presenting this phenomenon, building upon percolation and the Ising model. Our construction relies on the introduction of a feedback mechanism which forces the system towards a critical regime.
|
My PhD dissertation (in French):
|
The two following articles present "self-critical" variants of percolation and of the Ising model:
|
Some toy models of self-organized criticality in percolation
ALEA, Lat. Am. J. Probab. Math. Stat. 19 (2022), 367–416 |
|
|
A planar Ising model of self-organized criticality Probab. Theory Related Fields 180 (2021), 163-198 |
The following work considers a model of self-organized criticality build by Matthias Gorny, and studies what happens if the mean-field interaction is replaced with long-range interactions:
|
An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range Electron. J. Probab. 29 (2024), article no. 15, 1–27. |
|
Interacting random loops
The following article, written with Lorenzo Taggi, Matteo Quattropani and Alexandra Quitmann deals with models of interacting random loops which are connected with several models of interest in statistical mechanics, such as the spin O(n) model, the Bose gas or the double dimer model.
|
Coexistence, enhancements and short loops in random walk loop soups
Probab. Math. Phys. 5 (2024), no. 3, 753–784 |