## Planning

## 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.*

## Supports de conférences

$\bullet$
Aurélie Chapron : *
Voronoi diagram on Riemannian manifolds.*

$\bullet$
Davide Giraudo : *
Hölderian weak invariance principle for strictly stationary sequences.*

$\bullet$
Livio Triolo : *
A stochastic model of metastatic proliferation.*

$\bullet$
Eva Locherbach : *
From Variable neighborhood random fields to the estimation of interaction graphs.
*

$\bullet$
Stefano Olla : *
Entropic hypercoercivity and hydrodynamic limits.
*

$\bullet$
Pierre Picco : *
Results on the random field Kac model and long-range Ising model in
one dimension done in collaboration with Enza Orlandi.
*

$\bullet$
Takahiro Nemoto : *
Population dynamics algorithms to evaluate large deviation functions: how to solve sampling issues.*

## Résumés des exposés

## 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 $\rho$.