I am a probabilist, and my research deals with several models of statistical mechanics.
Self-Organized Criticality
I completed my PhD under the supervision of Raphaël Cerf (ENS Paris / Université Paris-Saclay) about self-organized criticality. This concept aims at describing some physical systems which present a “critical” behaviour without having to tune a parameter like temperature to a critical value.
Building upon percolation and the Ising model, we defined several toy models presenting this phenomenon, by introducing a feedback mechanism which forces the system towards a critical regime.
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Some toy models of self-organized criticality in percolation
with Raphaël Cerf ALEA, Lat. Am. J. Probab. Math. Stat. 19 (2022), 367–416 doi.org/10.30757/ALEA.v19-14 arxiv.org/abs/1912.06639 |
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A planar Ising model of self-organized criticality
Probab. Theory Related Fields 180 (2021), 163-198 doi.org/10.1007/s00440-021-01025-9 arxiv.org/abs/2002.08337 |
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An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range
Preprint (2021) (was previously published on the arXiv in two separate papers which were eventually merged due to a simplification) arxiv.org/abs/2110.07949 |
Activated Random Walks
Consider, on each site of a graph, for example the infinite lattice $\smash{\mathbb{Z}^d}$, a certain number of frogs (or particles) which can be either active or sleeping. We start with a random configuration of active frogs, with for example i.i.d. numbers of active frogs on each site, with mean $\smash{\mu}$ (the precise law does not matter much). Each active frog performs a continuous-time random walk on $\smash{\mathbb{Z}^d}$, with jump rate 1. Besides, when an active frog is alone on a site, it falls asleep with a certain rate $\smash{\lambda}$. Sleeping frogs stop moving, until another frog arrives on the same site, which turns it back into the active state.
It turns out that there exists a critical density $\smash{\mu_c(\lambda)}$, which depends on the sleep rate $\smash{\lambda}$, below which each frog almost surely eventually falls asleep forever, and above which each frog almost surely walks an infinite number or steps.
This model has drawn some attention these last years, especially for its connection with self-organized criticality. For example, starting with many active frogs on a single site, one expects that the frogs eventually stabilize in a ball with a critical density of sleeping frogs inside.
In my works with Alexandre Gaudillière and Amine Asselah, we studied the phase transition of this model and proved the existence of a non-trivial active phase in dimension 2.
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Active Phase for Activated Random Walks on the Lattice in all Dimensions
with Alexandre Gaudillière to appear in Annales de l'Institut Henri Poincaré arxiv.org/abs/2203.02476 |
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The Critical Density for Activated Random Walks is always less than 1
with Amine Asselah and Alexandre Gaudillière Preprint (2022) arxiv.org/abs/2210.04779 |
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Random loops and Spin-$O(n)$ model
With Lorenzo Taggi, Matteo Quattropani (Sapienza Università di Roma) and Alexandra Quitmann (WIAS Berlin), I am studying random loop models which are connected with the Spin-$\smash{O(n)}$ model, which is a generalization of the Ising model with spins taking values in a sphere of dimension $\smash{n-1}$.