15 et 16 septembre 2016 - Université de Rouen

## Jeudi 15 septembre

10h45 : Ouverture officielle
11h00 - 12h30 : N. Dirr - Recent developments in scaling limits for Glauber evolution with Kac Potentials (Part I).
12h30 - 14h00 : Déjeuner
14h00 - 15h45 : E. Locherbach - From Variable neighborhood random fields to the estimation of interaction graphs.
14h45 - 15h30 : L. Triolo - A stochastic model of metastatic proliferation.
15h30 - 16h15 : C. Külske - The Widom-Rowlinson model under spin ip: Immediate loss and sharp recovery of quasi-locality.
16h15 - 16h30 : Pause café
16h30 - 17h15 : M. Jara - Non-equilibrium fluctuations of one-dimensional particle systems.
17h15 - 18h15 : N. Dirr - Recent developments in scaling limits for Glauber evolution with Kac Potentials (Part II).

## Vendredi 16 septembre

9h00 - 9h45 : P. Picco - Results on the random field Kac model and long-range Ising model in one dimension done in collaboration with Enza Orlandi.
9h45 - 10h30 : S. Olla - Entropic hypercoercivity and hydrodynamic limits.
10h30 - 10h45 : Pause café
10h45 - 12h00 : N. Dirr - Recent developments in scaling limits for Glauber evolution with Kac Potentials (Part III).
12h00 - 13h30 : Déjeuner
13h30 - 14h15 : T. Nemoto - Population dynamics algorithms to evaluate large deviation functions: how to solve sampling issues.
14h15 - 15h00 : D. Giraudo - Hölderian weak invariance principle for strictly stationary sequences.
15h00 - 15h15 : Pause café
15h15 - 16h00 : A. Chapron - Voronoi diagram on Riemannian manifolds.

## Mini-cours

Nicolas Dirr (Cardiff University, United Kingdom) Recent developments in scaling limits for Glauber evolution with Kac Potentials 20 years ago, Anna De Masi. Enza Orlandi, Errico Presutti and Livio Triolo published a series of papers on scaling limits for Glauber evolution and Kac potential which had a profound impact on the field. In my series of talks I will review some of the developments since then, focussing on the following topics:
- Convergence to a sharp interface limit: Alternative proofs, extensions and recent applications;
- Connection with generalized solutions for PDEs, the use of the comparison principle and the questions of continuation beyond singularities;
- Large deviations, optimal transition paths, and the emergence of a (pseudo)metric structure related to the particle model.

## Conférences

Aurélie Chapron (Université Paris Ouest Nanterre et Université de Rouen) Voronoi diagram on Riemannian manifolds. We examine the Voronoï diagram generated by a homogeneous Poisson point process on a Riemannian manifold. More precisely, we show a link between the mean characteristics of a cell and the scalar curvature of the manifold. We give a high intensity asymptotic expansion of the mean number of vertices. We start with the two-dimensional case and explain how to extend the result in higher dimension.The proof relies notably on classical comparison results from Riemannian geometry and on a general change of variables formula of Blaschke-Petkantschin type. We then apply this result to give a probabilistic proof of the Gauss-Bonnet theorem in dimension 2.

Davide Giraudo (Université de Rouen) Hölderian weak invariance principle for strictly stationary sequences. The understanding of the asymptotic behaviour of partial sums of strictly stationary sequences is an important problem in probability theory. Defining the random function $S_n^{\mathrm{pl}}(f,\cdot)$ by \begin{equation*} S_n^{\mathrm{pl}}(f,t)=\begin{cases} S_k(f)=\sum_{i=0}^{k-1}f\circ T^i &\mbox{ if }t=k/n, 0\leqslant k\leqslant n;\\ \mbox{linear interpolation}& \mbox{ if }t\in \left(k/n,(k+1)/n\right), \end{cases} \end{equation*} we may study the convergence in distribution of the sequence $\left(n^{-1/2}S_n^{\mathrm{pl}}(f,\cdot)\right)_{n\geqslant 1}$ in some function spaces. The case of Hölder space of exponent smaller than $1/2$ seems to benefit less attention than the space of continuous function. The case of i.i.d. sequence has been addressed by Rackauskas and Suquet (2003). In this talk, we shall present results obtained by martingale approximation.

Milton Jara (IMPA, Rio de Janeiro) Non-equilibrium fluctuations of one-dimensional particle systems. We present a version of the second-order Boltzmann-Gibbs principle introduced by Gonçalves-J. for one-dimensional, conservative particle systems starting from non-equilibrium initial configurations. As an application, we prove that the entropy production rate in Yau's relative entropy method is bounded by a constant independent of the size of the lattice. In particular, this bounds allows to prove convergence of the fluctuations of the density around the hydrodynamic limit to a Gaussian process.
Joint work with Otávio Menezes (IMPA).

Christof Külske (RUHR-Universitât, Bochum) The Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasi-locality. We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides the first analysis of a Gibbs-non Gibbs transition for point particles under time-evolution, which previously has been done only for lattice systems. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices.
Joint work with Benedikt Jahnel (Wias, Berlin).

Eva Löcherbach (Université de Cergy-Pontoise) From Variable neighborhood random fields to the estimation of interaction graphs. In 2011, we have published a paper together with Enza Orlandi devoted to the study of random fields having interaction neighborhoods of variable "range" - the main idea being that the actual range of interaction might depend on a neighborhood that changes with the boundary conditions. This is a natural extension of the notion of variable-length Markov chains introduced by Rissanen (1983) in his classical paper to the frame of random fields. In the 2011-paper, we have proposed an estimator to retrieve the radius of the smallest ball containing the context based on a realization of the field. This estimator did not provide further information about the structure of interactions within the estimated ball. In a recent study, together with Aline Duarte, Antonio Galves and Guilherme Ost, we have extended this statistical procedure to estimate the interaction graph in systems of interacting neurons.
In my talk I will try to give an overview over the achieved results and explain in detail the notion of "variable neighborhood random fields" and the structure of the two estimators. I will also discuss important probabilistic tools that allow to assess the quality of the proposed estimators.

Takahiro Nemoto (Universités Paris Diderot et Pierre et Marie Curie) Population dynamics algorithms to evaluate large deviation functions: how to solve sampling issues. Rare events play an important role in many physical, chemical and biological phenomena. Examples are observed at small scales, e.g. in homogeneous nucleation, catalysis reactions and protein folding, and, at large scales, in geometrical reversal, rogue waves or the chaotic motion in solar system, among others. Many methods have been developed in order to characterise those rare events. In mathematics, one of the main methods to describe rare events are large deviation principles, in which one focuses on events deviated from the law of large numbers in a large-system or in a large-time limit. Their distribution is described by a large deviation function.
Since rare events are hardly observed by definition, the question of how to accelerate the measurement of those events in numerical simulations is an important topic. It is known as rare event samplings. For large deviations, a method based on "selection-mutation of clones" known as a population dynamics has been developed [1]. In this method, several copies of the system are simulated at the same time, and during these simulations, the "better clones" showing the desired rare behaviour are selected and multiplied repeatedly, while the others are eliminated.
In this seminar, we discuss this population dynamics algorithm for evaluating large deviations of additive observables in Markov processes. This method exhibits systematic errors which can be large in some circumstances, particularly for systems with many degrees of freedom, or close to dynamical phase transitions, or also in a weak noise asymptotics. We explain the origin of these errors, and show how they can be mitigated by introducing control forces within the algorithm [2]. These forces are determined by an iteration-and-feedback scheme, inspired by multicanonical methods in equilibrium sampling. We demonstrate substantially improved results in a simple model and we discuss potential applications to more complex systems.

[1] C. Giardinà, J. Kurchan, L. Peliti, Phys. Rev. Lett. 96, 120603 (2006)
[2] T. Takahiro, F. Bouchet, R. Jack and V. Lecomte, Phys. Rev. E 93, 062123 (2016)

Stefano Olla (Université Paris Dauphine) Entropic hypercoercivity and hydrodynamic limits. Entropic hypercoercivity provides estimates uniform in the dimensions of the dynamics, that are useful in proving hydrodynamic limit. In particular we use it to prove diffusive isothermal macroscopic transformations for a chain of non-linear oscillators immersed in a heat bath with a gradient of temperatures.

Pierre Picco (Institut de Mathématiques de Marseille) Results on the random field Kac model and long-range Ising model in one dimension done in collaboration with Enza Orlandi. With Enza Orlandi we worked on equilibrium properties of the "simplest" disordered models which are statistical mechanics of Ising systems with external random magnetic fields.
I will review the four articles we did on the Kac model and the two we did on long range model. Some open problems we had no time to solve will be mentionned.

Livio Triolo  (Università di Roma Tor Vergata) A stochastic model of metastatic proliferation Starting from a deterministic model by Iwata, Kawasaki and Shigesada, K. Ravishankar and I consider the process of metastatization as a collection of independent birth-death processes describing the evolution of tumours in the size-space, together with a suitable production term associated to the metastatic proliferation. We directly write the evolution for the distribution $\rho$ and consider how changes in the parameters affect the qualitative behavior of $\rho$.