Nicolas Forien

I am associate professor ("maître de conférences") at Ceremade, the mathematics department of Université Paris-Dauphine. I am a member of the research group "Probability and Statistics".



Year 2023-2024:


My research work deals with several discrete probability models originating from statistical mechanics, such as activated random walks, sandpile models, spin models or percolation.

I completed my PhD at ENS Paris, under the supervision of Raphaël Cerf, about self-organized criticality. Then, I was a temporary teaching and research assistant ("attaché temporaire d'enseignement et de recherche") at Aix-Marseille Université, where I worked on activated random walks, with Alexandre Gaudillière and Amine Asselah. After this, I was a postdoc student in the university "La Sapienza" in Rome, where I studied models of interacting random loops, spin models and sanpile models, with Lorenzo Taggi, Matteo Quattropani, Alexandra Quitmann and Concetta Campailla. My CV is available here.

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.

During my PhD thesis with Raphaël Cerf, we defined several toy models presenting this phenomenon, building upon percolation and the Ising model. More precisely, our construction relies on the introduction of a feedback mechanism which forces the system towards a critical regime.

Link to my PhD dissertation (in French):
« Autour de la criticité auto-organisée »
(Around self-organized criticality)

Some toy models of self-organized criticality in percolation
with Raphaël Cerf

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

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.

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.

Active Phase for Activated Random Walks on the Lattice in all Dimensions
avec Alexandre Gaudillière

to appear in Annales de l'Institut Henri Poincaré

The Critical Density for Activated Random Walks is always less than 1
with Amine Asselah and Alexandre Gaudillière

Preprint (2022)

Interacting random loops

With Lorenzo Taggi, Matteo Quattropani and Alexandra Quitmann, I am studying models of interacting random loops which are connected with several models of interest in statistical mechanics, such as the spin-$\smash{O(n)}$ model, the Bose gas or the double dimer model.

We are interested in various questions relating to this family of models, in particular the existence of macroscopic loops, i.e. of a length proportional to the size of the graph when the latter tends towards infinity.

Coexistence, enhancements and short loops in random walk loop soups
with Matteo Quattropani, Alexandra Quitmann and Lorenzo Taggi

Preprint (2023)