# Researches

I am interested in probability theory, particularly in models of statistical mechanics which exhibit the phenomenon of self-organized criticality. This phenomenon characterizes complex physical systems, composed of a large number of dynamically interacting elements, which are naturally attracted by critical points, without any external intervention. During my PhD Thesis, I modified the mean-field version of the Ising model in order to design and study a model having this behavior.

I made my thesis, entitled A Curie-Weiss Model of Self-Organized Criticality, at Université Paris-Sud and ENS Paris, under the supervision of Raphaël Cerf. I defended my thesis on 8 June 2015.

After my thesis, I spent the following two years as Lecteur Hadamard (post-doc of FMJH) at Université Paris-Sud. While continuing to study the model of my thesis, I also studied the geometric properties (perimeter, volume, exploration time) of a critical percolation cluster on the UIPT (Uniform Infinite Planar Triangulation).

## Thesis

• My PhD Thesis : A Curie-Weiss Model of Self-Organized Criticality
Abstract

In their famous 1987 article, Per Bak, Chao Tang and Kurt Wiesenfeld showed that certain complex systems, composed of a large number of dynamically interacting elements, are naturally attracted by critical points, without any external intervention. This phenomenon, called self-organized criticality, can be observed empirically or simulated on a computer in various models. However the mathematical analysis of these models turns out to be extremely difficult. Even models whose definition seems simple, such as the models describing the dynamics of a sandpile, are not well understood mathematically. The goal of this thesis is to design a model exhibiting self-organized criticality, which is as simple as possible, and which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. For a class of symmetric distributions whose density satisfies some integrability conditions, we prove that the sum $S_{n}$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model: the fluctuations are of order $n^{3/4}$ and the limiting law is $C \exp(-\lambda x^{4})\,dx$ where $C$ and $\lambda$ are suitable positive constants. Our study led us to generalize this model in several directions: the multidimensional case, more general interacting functions, extension to self-interactions leading to fluctuations with order $n^{5/6}$. We also study dynamic models whose invariant distribution is the law of our Curie-Weiss model of self-organized criticality. (Official version available here)

• My Master Thesis : The Curie-Weiss Model and its Generalizations

## Publications

• Path-space moderate deviations for a Curie-Weiss model of Self-Organized Criticality: the Gaussian case,
Francesca Collet, Matthias Gorny & Richard C. Kraaij
to appear in Annales de l'Institut Henri Poincaré, 2019
Abstract

We analyze the dynamics of moderate fluctuations for the magnetization of the Curie-Weiss model of self-organized criticality introduced in (Gorny, 2017). We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations.

• The geometry of a critical percolation cluster on the UIPT, Matthias Gorny, Édouard Maurel-Segala & Arvind Singh
Annales de l'Institut Henri Poincaré, 54(4):2203-2238, 2018.
Abstract

We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical cluster. The exponents obtained here differs by a factor 2 from those computed previously in (Angel, Curien. 2015) in the case of critical site percolation on the uniform infinite half-plane triangulation.

• The Curie-Weiss Model of SOC in Higher Dimension, Matthias Gorny
Annales de la Faculté des Sciences de Toulouse, 28(1):91-108, 2019.
Abstract

We build and study a multidimensional version of the Curie-Weiss model of self-organized criticality we have designed in (Cerf, Gorny. 2016). For symmetric distributions satisfying some integrability condition, we prove that the sum $S_n$ of the randoms vectors in the model has a typical critical asymptotic behaviour. The fluctuations are of order $n^{3/4}$ and the limiting law has a density proportional to the exponential of a fourth-degree polynomial.

• A Dynamical Curie-Weiss Model of SOC: The Gaussian Case, Matthias Gorny
Annales de l'Institut Henri Poincaré, 53(2):658-678, 2017.
Abstract

In this paper, we introduce a Markov process whose unique invariant distribution is the Curie-Weiss model of self-organized criticality (SOC) we designed and studied in (Cerf, Gorny. 2016). In the Gaussian case, we prove rigorously that it is a dynamical model of SOC: the fluctuations of the sum $S_{n}(\,\cdot\,)$ of the process evolve in a time scale of order $\sqrt{n}$ and in a space scale of order $n^{3/4}$ and the limiting process is the solution of a critical'' stochastic differential equation.

• Fluctuations of the Self-Normalized Sum in the Curie-Weiss Model of SOC, Matthias Gorny & S.R.S Varadhan
Journal of Statistical Physics, 160(3):513-518, 2015.
Abstract

We extend the main theorem of (Cerf, Gorny. 2016) about the fluctuations in the Curie-Weiss model of SOC in the symmetric case. We present a short proof using the Hubbard-Stratonovich transformation with the self-normalized sum of the random variables.

• An Exponential Inequality for Symmetric Random variables, Raphaël Cerf & Matthias Gorny
American Mathematical Monthly, 122(8):786-789, 2015.
Abstract

We prove a simple exponential inequality which gives a control on the first two empirical moments of a sequence of independent identically distributed symmetric real-valued random variables. Let $n\geq 1$ and let $X_1,\dots,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have $\mathbb{P}\big({X_1+\dots+X_n}\geq x,\,{X_1^2+\dots+X_n^2}\leq y\big)< \exp\left(-\frac{x^2}{2y}\right)\,.$

• A Lower Bound on the Relative Entropy with Respect to a Symmetric Probability, Raphaël Cerf & Matthias Gorny
Electronic Communications in Probability, 20(5):1-5, 2015.
Abstract

Let $\rho$ and $\mu$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(\mu|\rho)$ the relative entropy of $\mu$ with respect to $\rho$. We prove that, if $\rho$ is symmetric and $\mu$ has a finite first moment, then $H(\mu|\rho)\geq \frac{\displaystyle{\left(\int_{\mathbb{R}}z\,d\mu(z)\right)^2}}{\displaystyle{2\int_{\mathbb{R}}z^2\,d\mu(z)}}\,,$ with equality if and only if $\mu=\rho$. We give an applicaion to the Curie-Weiss model of self-organized criticality.

• A Self-Interaction Leading to Fluctuations of Order $n^{5/6}$, Matthias Gorny
Markov Processes and Related Fields, 21:205-248, 2015.
Abstract

In (Cerf, Gorny. 2016), we built and studied a Curie-Weiss model exhibiting self-organized criticality: it is a model with a self-interaction leading to fluctuations of order $n^{3/4}$ and a limiting law proportional to $\exp(-x^4/12)$. In this paper we modify our model in order to kill the term $x^4$'' and to obtain a self-interaction leading to fluctuations of order $n^{5/6}$ and a limiting law $C\,\exp(-\lambda x^6)\,dx$, for suitable positive constants $C$ and $\lambda$.

• The Cramér Condition for the Curie-Weiss Model of SOC, Matthias Gorny
Brazilian Journal of Probability and Statistics, 30(3):401-431, 2016.
Abstract

We pursue the study of the Curie-Weiss model of self-organized criticality we designed in (Cerf, Gorny. 2016). We extend our results to more general interaction functions and we prove that, for a class of symmetric distributions satisfying a Cramér condition $(C)$ and some integrability hypothesis, the sum $S_{n}$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order $n^{3/4}$ and the limiting law is $k \exp(-\lambda x^{4})\,dx$ where $k$ and $\lambda$ are suitable positive constants. In (Cerf, Gorny. 2016) we obtained these results only for distributions having an even density.

• A Curie-Weiss Model of Self-Organized Criticality: The Gaussian Case, Matthias Gorny
Markov Processes and Related Fields, 20(3):563-576, 2014.
Abstract

We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. With the help of exact computations, we show that, in the case of a centered Gaussian measure with positive variance $\sigma^{2}$, the sum $S_n$ of the random variables has fluctuations of order $n^{3/4}$ and that $S_n/n^{3/4}$ converges to the distribution $C \exp(-x^{4}/(4\sigma^4))\,dx$ where $C$ is a suitable positive constant.

• A Curie-Weiss Model of Self-Organized Criticality, Raphaël Cerf & Matthias Gorny
The Annals of Probability, 44(1):444-478, 2016.
Abstract

We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. For a class of symmetric distributions whose density satisfies some integrability conditions, we prove that the sum $S_{n}$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order $n^{3/4}$ and the limiting law is $C \exp(-\lambda x^{4})\,dx$ where $C$ and $\lambda$ are suitable positive constants.