# Random media, phase transitions

# and information theory

## September 8th – 19th, 2008

## Talks

Talks will be 45 mn long.

**Enrique Andjel (Marseille): **
The one-dimensional contact process with two types.

*Abstract : *
We consider a Markov process on $\{0,1,2\}^Z$
whose transition rates for coordinate $x$ and configuration
$\eta$ are given by: $1\rightarrow 0$ at rate 1, $2\rightarrow 0$
at rate 1, $0\rightarrow 1$ at rate
$\lambda_1(1_{\eta(x-1)=1}+1_{\eta(x+1)=1})$ and $0\rightarrow 2$
at rate $\lambda_2(1_{\eta(x-1)=2}+1_{\eta(x+1)=2})$. We show
that if an initial configuration has a single "2", then the
population of "2"'s has a positive probability of surviving
forever if there are only finite many "1"'s either to the left or
to the right of the unique "2", and $\lambda_2$ is bigger than
the critical value of the contact process and than
$\lambda_1-\epsilon$ for some $\epsilon>0$. Moreover, this does
not happen if $\lambda_1=\lambda_2$ and the initial condition has
infinitely many "1" both to the left and to the right of the
unique "2".

**Amine Asselah (Paris): **
Large deviation estimates for the energy of a
randomly charged random polymer.

*Abstract : *
We consider the energy of randomly charged random walk.
The random walk evolves on $\Z^d$ with $d\ge 3$, and the
charges satisfy Cramer's condition.
We study the probability, averaged over both randomness,
that the energy is large.

**Christophe Bahadoran (Clermont-Ferrand): **
The quasi-potential and a generalized Derrida-Lebowitz-Speer functional
for asymmetric particle systems with open boundaries.

*Abstract : *
Stationary large deviation functionals for ASEP or SSEP with open boundaries
were derived by Derrida, Lebowitz and Speer by explicit computations. Their
non-local structure reflects long-range correlations in nonequilibrium steady
states. For SSEP a different approach was initiated by Bertini et al., the
so-called ``macroscopic fluctuation theory", which derives the stationary
functional as a quasi-potential associated to a dynamical one. This approach
partially extends to other models but was so far restricted to diffusive
models. I will explain how it can be made effective for TASEP and more
general asymmetric particle systems. The outcome is a generalization of the DLS
functional and identification of the mimimizing paths. In our case the
dynamical functional is quite different because of shocks (including the
boundaries). It consists of a bulk term introduced by Jensen and Varadhan
(which measures how far a weak solution of Burger's equation is from being
entropic) and a boundary term introduced by Bodineau and Derrida (which
measures violation of boundary conditions in the sense introduced by Bardos,
Leroux and Nédélec for hyperbolic conservation laws). As opposed to the
diffusive case the variational problem cannot be solved by a Hamilton-Jacobi
equation.

**Raffaele Esposito (L'Aquila): **
Critical Droplet Size for the Mesoscopic Kac-Ising Model.

*Abstract : *
The equilibrium states of the Kac-Ising model in a box of size
L are described, on a mesoscopic scale, by magnetization profiles which
are the constrained minimizers of the Gates-Penrose free energy functional
with fixed total magnetization. When the total magnetization is slightly
above the negative minimum, the optimal profile may or may not create a
droplet of positive magnetization, depending on the size of the possible
droplet as function of L. There is a critical power of growth with L, and
a corresponding critical size, below which the droplet evaporates and
above which a droplet of spherical shape is created. A similar behavior
has been stated by Biskup Chayes and Kotecky for the two dimensional local
Ising model via a microscopic analysis of contours. Here, extending a
previous result on the Allen-Cahn free energy functional, we prove the
existence of the above droplet size by suitable upper and lower bounds for
the Gates-Penrose free energy functional by using rearrangement
techniques. The results give information on the structure of the
microscopic states of the Kac-Ising model by means of large deviations
arguments.

**Antonio Galves (São Paulo): **
Stochastic chains with memory of variable length: perfect simulation
and consequences.

*Abstract : *
Stochastic chains with memory of variable length constitute an
interesting family of stochastic chains of infinite order on a finite
alphabet. The idea is that for each past, only a finite suffix of the
past, called context, is enough to predict the next symbol. These
models were first introduced in the information theory literature by
Rissanen (1983) as a universal tool to perform data
compression. Subsequently, they have been used to model up scientific
data in areas as different as biology, linguistics and music.

In recent years chains with memory of variable length received a lot
of attention in the statistics literature, with several papers
dedicated to the study of the properties of the algorithm Context and
other estimators of the probabilistic context tree defining the chain.
But not much has been done to better understand the probabilistic
structure of these chains. Not even the basic problem of the existence
of these chains when the tree of contexts is unbounded has been
properly addressed.

In my talk I will present a new way to perform a perfect simulation of
a chain with memory of variable length. This will imply the existence
and uniqueness of the stationary chain. This will also provide an
upper bound for the rate at which the chain converges to the
stationary regime. The success of the procedure is assured by a new
type of condition on the rate at which the length of the contexts
grow.

**Thierry Gobron (Cergy): **
Attractiveness and couplings for conservative particle systems.

*Abstract : *
Attractiveness is a fundamental tool in the study of interacting particle
systems. On classical models such a simple exclusion, this property is shown
to hold through the basic coupling construction, which proves the existence of
a markovian coupled process $(\xi_t,zeta_t)_{t\geq 0}$ that satisfies:(P) for
any two initial configurations $\xi_0\leq\zeta_0$ (coordinate-wise),
$\xi_t\leq\zeta_t$ a.s. for all $t\geq 0$.
We generalize this classical result on two classes of models, on which the
basic coupling construction is however not possible:In one part, we consider
conservative particle systems on $\Z^d$ for which, in each transition, $k$
particles can jump between sites, with $1\leq k$. In the second part, we
consider exclusion systems with interaction. In both cases, we give necessary
and sufficient conditions on the rates under which those systems are
attractive, and give some details on the construction of a markovian coupled
process satisfying (P).We also emphasize some of the main differences between
basic coupling and the present construction.

**Dima Ioffe (Haifa): **
Semi-directed polymers at weak disorder.

*Abstract : *
We show that semi-directed (or crossing) polymers in dimensions larger
than three are diffusive at weak disorder.

**Vlada Limic (Marseille): **
The Lambda-coalescent speed of coming down from infinity.

*Abstract : *
Consider a $\Lambda$-coalescent that
comes down from infinity (meaning that
it starts from a configuration containing
infinitely many blocks at time $0$, yet
it has a finite number $N_t$ of blocks at any positive time
$t>0$). We
exhibit a deterministic function $v:(0,\infty)\to (0,\infty)$,
such that $N_t/v(t)\to 1$, almost surely and in $L^p$ for any $p\geq 1$, as $t\to 0$.
Our approach relies on a novel martingale technique.

**Kirone Mallick (Saclay): **
Exact results for the Asymmetric Exclusion Process.

*Abstract : *
The Asymmetric Simple Exclusion Process (ASEP) plays the role of a
paradigm in Non-Equilibrium Statistical Mechanics: it is one of the
simplest interacting N-body systems far from equilibrium that can
be solved analytically. By using the Bethe Ansatz, we calculate the
spectral gap of the model and predict global spectral properties
such as the existence of multiplets. We then discuss the
fluctuations of the current in the stationary state. Finally, we
explain that the stationary state of the ASEP and of some of its
generalizations with multiple classes of particles has an
underlying combinatorial structure that leads naturally to a
matrix product representation.

**Rossana Marra (Roma): **
Segregation phase transition.

*Abstract : *
A system of two species of particles interacting through a long-range
(Kac) potential, repulsive between different species is modeled by
interacting Ornstein-Uhlenbeck processes. This system undergoes a
first-order phase transition with coexistence at low temperature of two
equilibrium states, one richer in species 1 and the other richer in
species 2. In the mean field limit the behavior of the system is
described by mesoscopic equations, called Vlasov-Fokker-Plank equations.
These equations have homogeneous and non-homogeneous stationary solutions
(like solitons in 1d), because of the phase transition.
I will discuss the stability of these solutions in 1d, proving the
stability of the constant one at high temperature and the stability of the
soliton at low temperature and the extension of this results to a finite
volume.

**Pierre Mathieu (Marseille): **
Entropy and rate of escape of random walks on groups.

*Abstract : *
We shall explain how the entropy of a random walk on a countable
group
can be interpreted as its rate of escape in an appropriate metric. We shall
also discuss
the connection with the 'entropy-rate of escape-log volume growth' inequality.

Reference: Annals Probability 2008.

**Eugene Pechersky (Moscou): **
Boolean percolation of Gibbs particle fields in
$R^2$.

*Abstract : *
The field is defined by a pair potential as Gibbs
reconstruction of Poisson field with its parameter $\lambda$. We
describe a region in plane $(\beta,\lambda)$ ($\beta$ is inverse
temperature) where an infinite cluster exists with probability 1.

**Christian P. Robert (Paris): **
Adaptive Importance Sampling in General Mixture Classes.

*Abstract : *
In this talk, we propose an adaptive algorithm that iteratively updates both
the weights and component parameters of a mixture importance sampling density
so as to optimise the importance sampling performances, as measured by an
entropy criterion. The method is shown to be applicable to a wide class of
importance sampling densities, which includes in particular mixtures of
multivariate Student t distributions. The performances of the proposed scheme
are studied on both artificial and real examples, highlighting in particular
the benefit of a novel Rao-Blackwellisation device which can be easily
incorporated in the updating scheme.

*Support : * ANR Adap'MC & ANR Ecosstat.

**Rinaldo Schinazi (Colorado Springs): **
Spatial and non spatial stochastic models for immune response.

*Abstract : *
We consider a model in which every individual can give birth to an individual
of the same type or to a mutated individual. Each type lives a mean 1
exponential time and then all the individuals of a given type are killed
simultaneously.
We compare a non spatial version of this model to versions on the square
lattice
and on the homogeneous tree.

**Gordon Slade (Vancouver): **
Invasion percolation on a regular tree.

*Abstract : *
Invasion percolation is a natural stochastic growth model,
closely related to critical percolation and the incipient infinite
cluster. This talk will report on joint work with Omer Angel, Jesse
Goodman, and Frank den Hollander, in which a detailed analysis of
invasion percolation on a rooted regular tree is carried out.
An important structural property of the invasion percolation cluster
will be described, which shows that the invasion percolation cluster
is stochastically dominated by the incipient infinite cluster,
and which explains why the two processes are in fact globally
different, despite the fact that they are locally the same far
above the root.

**Yvan Velenik (Genève): **
Ballistic phase of self-interacting polymers.

*Abstract : *
We consider a general class of polymers with attractive or
repulsive self-interaction. Examples of repulsive models are the SAW and
the Domb-Joyce model; examples of attractive models are reinforced
polymers and annealed RW in random potential. We suppose that the
polymer is subject to a force pulling it at one end, the other end being
pinned. We prove that there is a sharp transition between a collapsed
and a ballistic phase in the attractive case as the force increases,
while the polymer is in the stretched phase for an arbitrarily weak
force in the repulsive case. We then provide a sharp description of the
ballistic regime (local limit theorem for the endpoint, Brownian
asymptotics, statistics of local observables, stability under
perturbations, etc.), and discuss properties of the phase transition
occurring in the attractive case.

## Posters

**Paul Chleboun (Warwick): **
Nonequilibrium phase transitions in perturbed particle systems.

*Abstract : *
We consider a one dimensional zero range process that is well known to exhibit
a condensation transition. We examine the effect of quenched disorder in the
particle interactions on the critical behaviour of the system. Recent
theoretical results on the change of phase diagram under perturbation are
supported by Monte Carlo simulations in the canonical ensemble. We also present
a numerical study of the stationary current for finite systems.

**Alexsandro Gallo (São Paulo): **
Markov approximation for unbounded variable length memory.

*Abstract : *
We consider chains with unbounded variable length memory. These chains
are a particular class of chains of infinite order, which are constructed using
a "probabilistic context tree". The subject of this poster is to show how we
can use these trees in order to obtain Markov approximation of infinite order
processes under new conditions.

**Pablo Martin Rodriguez (São Paulo): **
An upper bound for the critical probability of a long range percolation model on trees.

*Abstract : *
We consider a simple long range percolation model on infinite
graphs and discuss sufficient conditions for phase transition. In addition,
we study the model on spherically symmetric trees and obtain an upper bound
for the critical probability. As a consequence we obtain the asymptotic
behavior of this critical parameter on homogeneous trees.

**IHP - Centre Emile Borel - 11, rue Pierre et Marie Curie 75005 Paris - France**

This page is maintained by
Ellen Saada. Thank you for reporting any problem in reading it.

Last update : September 10, 2008.