# Interacting Particle Systems,

# Statistical Mechanics

# and Probability Theory

## September 5th – December 19th, 2008

## Scheduled courses

### Long courses

Those courses will be between 16 and 20 hours long.

**Pablo Ferrari (São Paulo) ** and **Ellen Saada (Rouen): **
Exclusion processes and applications.

*Length : * 2h per week, for 9 weeks.

*When : * From September 23th
to December 2nd, 2008, on Tuesdays, 13h30–15h30
(except the weeks of October 27th and November 10th).

*Pre-requisites : * Familiarity with probability theory,
in particular continuous time Markov chains.

*Abstract : *

Part 1: generalities on simple exclusion, shocks,
couplings, multiclass processes, oriented percolation and competition interface.

Part 2: Euler
hydrodynamics (in strong and weak senses)
of one-dimensional conservative attractive particle systems.
This part is based on the papers:

Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E. A constructive
approach to Euler hydrodynamics for attractive particle systems.
Application to k-step exclusion. * Stoch. Process. Appl.***
99** (2002) no. 1, 1-30.

Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E. Euler
hydrodynamics of one-dimensional attractive particle systems. *
Ann. Probab.* ** 34** (2006), 1339-1369.

Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E.
Strong hydrodynamic limit for attractive particle
systems on Z. Preprint (2008). [http://arxiv.org/abs/0804.2345]

**Geoffrey Grimmett (Cambridge): **
Discrete Spatial and Physical Processes in Probability.

*Included in : * Master 2
Probabilités et Statistiques, Orsay, and in Master 2
Probabilités, Paris 6.

*Length : * 3h per week, for 8 weeks.

*When : * From September 29th, 2008, on Wednesdays, 11h–12h30, 14h–15h30
(except the week of October 27th).

*Pre-requisites : * Familiarity with basic probability theory including
elements of discrete-time Markov chains, random walk, basic martingale
theory, and measure theory.

*Abstract : *
Many of the most beautiful and important problems of probability theory
involve random processes associated with networks. The inspirations for
such problems come often from areas of applied science stretching from
physics to epidemiology and biology, and their solutions feed back into
the field of origin.

The primary target of this lecture series will be to present a coherent
theory of discrete spatial processes emerging from a number of problem
areas including: random walk, random trees, percolation, models for
ferromagnets and spin glasses, and interacting particle systems. In each
case, we will progress from the basics to the principal open problems of
the relevant field. Special emphasis will be placed on connections between
topics, and on generic methodology including correlation inequalities and
concentration.

**Hermann Thorisson (Reykjavik): **
Coupling Methods in Probability Theory.

*Included in : * Master 2
Probabilités, Paris 6.

*Length : * 3h per week, for 7 weeks.

*When : * From September 29th, 2008, on Thursdays, 11h–12h30, 14h–15h30
(except the weeks of October 20th, October 27th).

*Pre-requisites : * (elementary) familiarity with measure theory, and
(elementary) familiarity with Markov Chains.

*Abstract : *
``Coupling has developed into one of the most powerful and beautiful tools
of probability theory. Beautiful (for probabilists) because it puts all
the emphasis on realizations and sample paths, rather than on measures;
beautiful also because a judicious coupling often makes self-evident the
proof of a previously difficult theorem." (From Andrew Barbour's review of
the book: Thorisson, H. (2000). Coupling, Stationarity, and Regeneration.
Springer, NY.)

Coupling means the joint construction of two or more random variables,
processes, or any random things. In these talks we shall first present the
method through a series of basic elementary examples such as maximal
coupling, Poisson approximation of a binomial variable; turning of
stochastic domination into pointwise domination, weak convergence into
pointwise convergence and liminf convergence of densities into pointwise
convergence where the random elements actually hit the limit; basic limit
theorem of Markov chains; and the celebrated coupling-from-the past
simulation algorithm.

We shall then outline a general coupling theory for stochastic processes
(Doeblin coupling, exact coupling, shift-coupling, and epsilon-couplings).
Applications to Markov Processes, Regenerative Processes and in Palm
Theory will be indicated. The view is then extended to random fields with
applications in Palm Theory, and finally to random elements under a
topological transformation group which opens up many new possibilities for
applications: self- similarity, exchangeability, rotational invariance,
the Lorentz and Poincar\'e transformations, ...

### Mini courses

Those courses will be between 4 and 10 hours long.

**Jean Bertoin (Paris): **
Second order reflection.

*Length : * 4 h.

*When : * Week of September 22th, 2008: Monday 22, 14h–16h and Friday 26, 10h–12h.

*Abstract : *
The first order reflection problem can be stated as follows:
given a continuous function $Y: [0,\infty[ \to \bbR$, find
a continuous function $X: [0,\infty[\to \bbR_+$ such that $X=Y+L$
where $L: [0,\infty[\to \bbR_+$ is a continuous non-decreasing
function which increases only when $X=0$.
It is well-known that it has a unique solution which is given by
the Skorohod's reflection principle:
$L_t= \sup\{Y_s^-: 0\le s \le t\}$, where $r_-$ denotes the negative part
of a real number $r$.

The second order reflection concerns smoother functions $Z$,
namely with a right-continuous derivative $\dot Z$.
Given such a function $Y: [0,\infty[ \to \bbR$, we look for
a smooth function $X: [0,\infty[\to \bbR_+$ such that
$\dot X=\dot Y+ A$
where $A: [0,\infty[\to \bbR_+$ is a non-decreasing
function such that $\dot A$ increases only when $X=0$.
Further, one requires a boundary condition for the reflection,
for instance $\dot X_t = -\dot X_{t-}$ when $X_t=0$
(the boundary is completely elastic), or
$\dot X_t = 0$ when $X_t=0$
(the boundary is totally inelastic).
It is known that such second order reflection problems have a unique
solution when $Y$ is analytic; however multiple solutions
may occur even in situations when $Y$ is of class ${\mathcal C}^{\infty}$.
Motivated by a question of Bertrand Maury,
we shall consider here this problem in the case when
$\dot Y$ is a standard Brownian motion (i.e. when $Y$ is a free Langevin
process).
We shall develop two different approaches, the first is based on
excursion theory for Markov processes, and the second on stochastic calculus.

**Erwin Bolthausen (Zürich): **
Perturbative theory for random walks in random
environments.

*Length : * 2h slots.

*When : * during the Fall School (week of September 8th, 2008).

*Abstract : *
e consider the classical model of a random walk in
a random environment, where the environment is given
by i.i.d. transition probabilities on a d-dimensional
lattice. For a fixed realization of the environment,
one considers a random walk with these transition
probabilities. Since the paper of Bricmont and
Kupiainen, it is known, that for small disorder,
the random walk remains diffusive in dimensions
larger than two, provided the law of the random
environment is symmetric under lattice isometries.
We present a new and simpler approach which uses
exit distributions. The mini-course may possibly
also cover the case of non-symmetric environment
distributions.

**Aernout van Enter (Groningen): **
Chaotic behaviour, metastates and randomness in
some simple models.

*Length : * 3 times 2 h.

*When : * during the Fall School (September 8th–19th, 2008).

*Abstract : *
We discuss the notions of chaotic size dependence, chaotic
temperature dependence, metastates, overlap distributions
and the role disorder plays for them to occur. We
illustrate
those on some simple lattice models.

References:

Aernout C. D. van Enter, Karel Netocny, Hendrikjan G.
Schaap: Incoherent boundary conditions and metastates
[http://front.math.ucdavis.edu/0502.4731]

Aernout C.D. van Enter:
On the set of pure states for some systems with non-periodic
long-range order. [http://www.ma.utexas.edu/mp_arc/c/96/96-93.ps.gz]

**Roberto Fernández (Rouen): **
Cluster expansions: overview and new convergence results.

*Length : * 5 lectures each 1.5 h.

*When : * week of November 17th, 2008, on Monday 17, 9h30–11h and 14h–15h30,
Tuesday 18, 9h30–11h and Friday 21, 9h30–11h
and 14h–15h30.

*Abstract : *
Introduced to describe low-density or high-temperature systems, cluster expansions have become the basic tool for rigorous perturbative arguments. The lectures will present an introduction to the technique, an overview of key applications, and a comparative review of available convergence criteria and their proofs.

Tentative program

I. Overview of cluster expansions

1.1 The general setting: hard-core, repulsive and stable systems

1.2 The issues: free energy and density expansions, correlations, mixing properties, zeros of partition functions

1.3 Graph theoretical framework

II. Review of applications in statistical mechanics and probability theory

2.1 Point processes and loss networks

2.2 Statistical mechanics: low- and high-temperature expansions of lattice gases and spin systems; duality

2.3 Geometrical models: FK model and its relations to Potts models and the chromatic polynomials

2.5 Grand canonical ensemble of continuous systems, hard spheres, stability

2.4 Characteristic functions of inhomogeneous Markov chains, central limit theorems

III. Convergence results and their proofs

3.1 Partitionability schemes and the Penrose identity; the tree-graph inequality

3.2 The ``classical'' convergence criterium based on tree-graph inequalities

3.3 Criteria based on Kirwood-Salzburg equations

3.3 Inductive proofs putting analyticity first

3.4 Improvement exploiting Penrose identity and the fix point property of of sums over trees

IV. Consequences and comparison

4.1 Systems with finite-coordination (domino, latticed gases in lattices)

4.2 Polymer or contour systems

4.3 Zeros of chromatic polynomials

4.4 Hard-sphere gas

**Olle Häggström (Göteborg): **
Coupling ideas in percolation theory.

*Length : * 5 times 45 mn.

*When : * during the Fall School (week of September 15th, 2008).

*Abstract : *
Coupling - the simultaneous contruction of two or more random objects on
the same probability space for the purpose of comparing or otherwise
drawing conclusions about their respective distributions - has turned out
in the last few decades to be a wonderfully fruitful probabilistic tools.
The main purpose of this lecture series is demonstrate some of the ways in
which the method has come to use in percolation and first-passage
percolation.

**Frank den Hollander (Leiden and Eindhoven): **
Metastability.

*Length : * 5 times 1.5 h.

*When : * week of December 15th, 2008: From Monday 15 to Friday 19, 9h30–11h00.

*Pre-requisites : * Basic knowledge of probability theory and
statistical physics, in particular, Markov chains and Gibbs
measures.

*Abstract : *
A physical, chemical or biological system driven by a noisy
microscopic dynamics may explore different regions of its state
space on different time scales: for certain values of the
interaction parameters the dynamics may move fast within
regions but slow between regions. The macroscopic phenomena
associated with this separation are collected under the name
metastability.

In this lecture series we consider two model systems from physics,
namely, Ising spins subject to a random spin-flip dynamics (which
is a model for magnetism) and lattice particles subject to a
random hopping dynamics (which is a model for condensation).
We consider various metastable regimes, compute the size and
the shape of so-called ``critical droplets'' for the metastable
transition, and identify the typical trajectories before and
after the transition.

The analyis is built on two approaches: pathwise (large deviations)
and potential-theoretic (electric networks).

Finally, we list what are the key challenges for the future.

**Fabio Martinelli (Roma): **
Kinetically constrained interacting particle systems.

*Length : * 4 lectures each 1.5 h.

*When : * week of November 3rd, 2008, on Monday 3, 9h30–11h and 14h–15h30,
Tuesday 4, 9h30–11h and Wednesday 5, 16h00–17h30.

*Abstract : *
Kinetically constrained spin models (KCSM) are interacting particle systems on a connected graph (e.g. $\bbZ^d$) with either Glauber-like or spin-exchange dynamics, reversible w.r.t. a simple product i.i.d Bernoulli($p$) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site or the jumping of a particle to a nearest neighbor empty site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. Such models have been intensively studied in the physical literature as simple systems sharing some of the basic features of a glass transition. Until recently a mathematical analysis of the relaxation to equilibrium of KCSM has been lacking. Here we will review recent results and techniques which provide a first rigorous framework for the above problems.

**Servet Martinez (Santiago): **
Quasi-stationary distributions (q.s.d.) in Markov Chains.

*Length : * 3 sessions, 4h. in total.

*When : * during the Fall School (September 8th–19th, 2008).

*Abstract : *
For (surely) killed Markov processes we study quasi-stationary distributions
(q.s.d.) that are those distributions that remain
invariant when the process is conditioned to survive. In a general framework we
supply the exponential time of absorption property when starting from a q.s.d.
and
we show that the absorption time and the state where it is absorbed
are independent random variables. For countable state spaces we give
the more general result on the existence of q.s.d.
[Ferrari, Kesten, Martinez, Picco]. The finite state case
serves to supply many of the ideas in the topic: quasi-limiting distributions,
ratio limit of survival probabilities
and the chain conditioned to survive at infinite.
For monotone processes
we compare the killing rate with the ergodic coefficient of convergence
to equilibrium.
We illustrate some of these concepts on the birth and death chains,
in particular the classification of the chain conditioned to survive at
infinite.
Further, we study the probabilistic evolution of a system of birth and death
chains with mutation, where the traits of individuals take values in
a compact metric space. In the evolution each one of the individuals can die or
generate a new individual,
the new element can have the same trait or it can mutate and its trait
is randomly distributed with a distribution depending on the parent.
The population is assumed to be extincted.
We give a recent result of [Collet, Meleard, Martinez, San Martin] that shows
that in an uniform case there exists a q.s.d.
such that when the process is conditioned to have a specific number of
traits and a specific number of individuals in each one
of this traits, the joint location of the traits is absolutely
continuous with respect to the product reference measure.

**Andrea Montanari (Stanford): **
Gibbs measures and phase transitions in information science.

*Length : * 3 or 4 sessions of 1.5h.

*When : * during the Fall School (September 8th–19th, 2008).

*Abstract : *
Theoretical models of disordered materials are the prototypes of a
large family of mathematical problems, ranging from theoretical
computer science (random combinatorial problems) to communications
(detection, decoding, estimation) and probability.
Such problems are currently the object of growing interest and
of an impressive convergence of different disciplines.

In all of these cases, one consider a joint distribution of a large number of
elementary degrees of freedom that is itself a random object.
Further, a graph can be often used to describe the locality structure
of interactions. A whole set of remarkable, but quite universal
prediction have been made by physicists, and are slowly being
confirmed rigorously. Among the others, the proliferation
of an exponential number of metastable states, or the occurrence of purely
dynamical phase transitions in reversible relaxational dynamics.

More in general, from a theoretical point of view, one is
interested in determining the typical properties of such random
distributions. For applications, one would like to define efficient
algorithms that allow to approximate the marginal distribution
of one (or a few) such degrees of freedom.
Among the most interesting such algorithms are message passing
algorithms (such as belief or survey propagation) and Monte
Carlo Markov Chain methods (the simplest example being Glauber
dynamics).

**Stefano Olla (Paris): **
Large deviations, thermodynamics and statistical mechanics.

*Included in : * Ecole doctorale EDIMMO.

*Length :* 3 lectures of
3h each.

*When : * On Mondays, September 22, September 29, October 13, 9h30–12h30.

*Abstract : *
This is an introductory course to the basic concepts of statistical
mechanics and thermodynamics, intended for graduate and PhD students.

Content:

A one-dimensional chain of anharmonic oscillators (of the Fermi-Pasta-Ulam
type) is the simple and non-trivial (non-linear) example to illustrate
statistical mechanics and thermodynamics. Large deviation for independent
random variables provides the mathematical tool.

1-The model, microcanonical, canonical, gran-canonical Gibbs measures.
Thermodynamic entropy, free energy, pressure,....

2. Large deviation for densities and the proof of the equivalence of
ensembles.

3. Dynamics, ergodicity, entropy production.

4. Non-equilibrium problems: Introduction to hydrodynamic limits.

**Yuval Peres (Seattle and Berkeley): **
Gravitational Allocation and Internal DLA.

*Length : * three lectures concentrated in one
week.

*When : * week of October 27th, 2008, on Tuesday 28, Thursday 30, Friday 31, 16h10–17h00.

*Abstract : *
Given a Poisson process of stars, Chandrasekhar (1942)
discovered that Newtonian gravitational force is convergent a.s. in dimension
at least 3, and has a stable law. We use this force (in the infinite
friction limit) to partition space into basins of attraction which
have equal volumes. We prove that the diameters of the basins have
exponential tail. (Joint work with S. Chatterjee, R. Peled,
D. Romik). In a separate work with L. Levine, we prove that internal
diffusion limited aggregation with multiple sources has a scaling
limit. This extends the pioneering work of Lawler, Bramson and Griffeath,
who in the 1990's analyzed internal DLA with a single source.
Both works are linked to a free boundary problem for the
Laplacian. The minicourse will be accessible to graduate students, and
I will develop the relevant background.

**Frank Redig (Leiden): **
Abelian sandpiles.

*Length : * 6h.

*When : * week of December 1st, 2008: Wednesday 3, Thursday 4, and Friday 5, 13h30–15h30.

*Abstract : *

Hour 1-2: Introduction to the sandpile model: motivation,
Dhar formalism of addition operators, burning algorithm,
connection with spanning trees, stationary probabilities
of weakly allowed subconfigurations (Bombay trick).

Hour 3-4: Infinite volume limits: basic questions. Infinite
volume in dimension 1, construction of stationary process,
given the existence of the infinite volume stationary measure.
Proof of existence of infinite volume limit of uniform measure
on recurrent configurations.

Hour 4-5: Organized versus self-organized criticality, sandpile
percolation and activated random walker models.
Perspectives and open questions.

**Fraydoun Rezakhanlou (Berkeley): **
Smoluchowski's equation and Gelation.

*Length :* 5 lectures, each 1 hour and 30 minutes long.

*When : * weeks of October 13th
and October 20th, 2008: Monday 13, Friday 17, Monday 20,
Thursday 23 and Friday 24, 14h–15h30.

*Abstract : *
Smoluchowski's equation describes the evolution of the cluster
densities for a model of coagulating-fragmening particles. It is
well-known that if the coagulation rate is supelinear in the cluster
sizes, then an instantaneous gelation occurs. This means that a particle
of infinite size is formed at time 0. This was proved by Ball and Carr in
the case
of the homogeneous Smoluchowski's equation and remains open in the
inhomogeneous case. There is a microscopic anolog of this result that is
studied by many researches for Makov processes with mean field
interactions.
In these talks, I will give an overview of the existing results in this
context.

**Senya Shlosman (Marseille): **
Statistical mechanics of queuing networks on highly-connected graphs.

*Length : * 5 lectures of 1.5 h each.

*When : * week of November 3rd, 2008, on Monday 3, Tuesday 4, 16h00–17h30,
Friday 7, 9h30–11h and 16h00–17h30.

*Abstract : *
We will study large network of servers, through which customers are
travelling, being served at different nodes of the network. We look for
global characteristics of the performance of the network. In particular, we
want to understand whether some sort of collective behavior can happen.

It turns out that the dynamical creation of long memory effects is possible
in certain networks. The nature of this long memory is similar to the phase
transition phenomenon in statistical physics. This phenomenon happens if the
load per server is high enough; otherwise the infinite network becomes
de-synchronized, no matter what interaction between servers is taking place.
The parameter of the load plays a role similar to the temperature in
statistical mechanics.The method of the study of this transition is based on
getting the detailed properties of certain infinite-dimensional dynamical
systems.

**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 : November 30, 2008.