# and Probability Theory

## Scheduled courses

### Long courses

Those courses will be between 16 and 20 hours long.

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]

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.

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.

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.

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.

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]

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

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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