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Matthias Gorny

Publications and source records attributed to Matthias Gorny.

11 recordsLinked to original sources
arXiv2018

Path-space moderate deviations for a Curie-Weiss model of self-organized criticality

The dynamical Curie-Weiss model of self-organized criticality (SOC) was introduced in \cite{Gor17} and it is derived from the classical generalized Curie-Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie-Weiss model of SOC [Cerf, Gorny 2016] as unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. 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. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical.

math.PR↗
arXiv2017

The geometry of a critical percolation cluster on the UIPT

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 by Angel and Curien (2015) in the case of critical site percolation on the uniform infinite half-plane triangulation.

math.PR↗
arXiv2016

A Curie-Weiss model of self-organized criticality

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 $C\exp(-λx^4)\,dx$ where $C$ and $λ$ are suitable positive constants.

math.PR↗
arXiv2015

A Dynamical Curie-Weiss Model of SOC: The Gaussian Case

In this paper, we introduce a Markov process whose unique invariant distribution is the Curie-Weiss model of self-organized criticality (SOC) we designed in arXiv:1301.6911. 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.

math.PR↗
arXiv2015

The Curie-Weiss Model of SOC in Higher Dimension

We build and study a multidimensional version of the Curie-Weiss model of self-organized criticality we have designed in arXiv:1301.6911. 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 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.

math.PR↗
arXiv2014

A Lower Bound on the Relative Entropy with Respect to a Symmetric Probability

Let $ρ$ and $μ$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(μ|ρ)$ the relative entropy of $μ$ with respect to $ρ$. We prove that, if $ρ$ is symmetric and $μ$ has a finite first moment, then \[ H(μ|ρ)\geq \frac{\displaystyle{(\int_{\mathbb{R}}z\,dμ(z))^2}}{\displaystyle{2\int_{\mathbb{R}}z^2\,dμ(z)}}\,,\] with equality if and only if $μ=ρ$.

math.PR↗
arXiv2014

An Exponential Inequality for Symmetric Random Variables

We prove the following exponential inequality: Let $n\geq 1$ and let $X_1,...,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have \[\mathbb{P}\big({X_1+...+X_n}\geq x,\, {X_1^2+...+X_n^2}\leq y\big)< \exp(-\frac{x^2}{2y})\,.\]

math.PR↗
arXiv2014

The Cramér Condition for the Curie-Weiss Model of SOC

We pursue the study of the Curie-Weiss model of self-organized criticality we designed in arXiv:1301.6911. 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 $k \exp(-λx^{4})\,dx$ where $k$ and $λ$ are suitable positive constants. In arXiv:1301.6911 we obtained these results only for distributions having an even density.

math.PR↗
arXiv2014

A Self-Interaction Leading to Fluctuations of Order $n^{5/6}$

In arXiv:1301.6911, 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 $C\,\exp(-λx^6)\,dx$, for suitable positive constants $C$ and $λ$.

math.PR↗
arXiv2013

A Curie-Weiss Model of Self-Organized Criticality : The Gaussian Case

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 $σ^{2}$, 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 $C \exp(-x^{4}/(4σ^4))\,dx$ where $C$ is a suitable positive constant.

math.PR↗