###Minicourses
Nalini Anantharaman (Université Strasbourg)
Yves Benoist (Université ParisSud, Orsay)
Philippe Bougerol (Université Pierre et Marie Curie, Paris)
JeanFrançois Quint (Université de Bordeaux)
###Talks
Richard Aoun (American University of Beyrouth)
Caroline Arvis (Université ParisSud, Orsay)
Emmanuel Breuillard (Université ParisSud, Orsay)
Reda Chhaibi (Université Paul Sabatier, Toulouse)
Anna Erschler (ENS, Paris)
Alex Gamburd (City University, New York)
Emmanuel Kowalski (ETH Zurich)
Pierre Mathieu (Université AixMarseille)
Amos Nevo (Technion Haifa)
Yehuda Shalom (University of Tel Aviv)
Peter Varju (University of Cambridge)
Andrzej Zuk (Université Diderot, Paris)







SCHEDULE 






Monday 



Tuesday 

Wednesday 

Thursday 

Friday 










































Welcome 



Quantum ergodicity I 

Random walks and representations 

Deviation inequalities and 

Random walks on discrete 








of compact groups II 

CLT for random walks on 

KdV equations 






















Random walks on linear 

Random walks on linear groups II: 

Random walks on linear groups III: 

Random walks on linear groups 

Invariant measures on aﬃne 



groups I: 







VI: 

Grassmannians 


Stationary 
measures 
and 

The central limit theorem 

Regularity of stationary measures 

Extensions, questions, and per 




Lyapunov exponents 






spectives 






















Random walks and represen 

Random walks on 

Quantum ergodicity II 

Isoperimetry, diﬀuse random 

Random walks and represen 



tations of compact groups I 

surfaces 



walks, and quotient of groups 

tations of compact groups 













III 















Lunch 



Lunch 

Lunch 

Lunch 

Lunch 

























Excursion 














The shapes 
of exponential 

Spectral gaps and random walks 


Uniform exponential escape of 





sums 







random matrix products from 












algebraic subgroups, and join 












ing of graphs 









Tee break 








Excursion 






New directions in Diophan 

Full dimension of Bernoulli convolu 


Random matrix products when 





tine approximation 


tions 



λ1 > λ2 








Excursion 







Functional 
equations 
in 










number theory and the re 











ßection principle for random 











walks 
















Conference cocktail 




















PDF to Word P2WConvertedByBCLTechnologies
These two lectures will be mostly devoted to defining and proving quantum ergodicity for laplacian eigenfunctions on discrete graphs. I will first recall the notion of quantum ergodicity for eigenfunctions on manifolds, and will state and prove the ”quantum ergodicity theorem” (due to Snirelman) in that setting : if the geodesic flow is ergodic, the laplacian eigenfunctions typically become equidistributed in phase space in the limit of large eigenvalues. I will then introduce the question of quantum ergodicity for finite discrete graphs whose size goes to infinity. With Etienne Le Masson, we proved a ”quantum ergodicity theorem” saying that if the graphs are regular, have large girth and are expanders, the eigenfunctions of the discrete laplacian are typically equidistributed, in a sense to be discussed. Recently, with Mostafa Sabri, we extended this theorem to nonregular graphs and more general discrete Schrdinger operators. _________________________________________________________________
R. AOUN: Random matrix products when λ_{1} > λ_{2}
In this talk, we present new results on random matrix products theory obtained in a joint work with Yves Guivarc’h. We consider a probability measure μ on the general linear group GL(V ) (on any local field) whose top Lyapunov exponent is simple and assume no other algebraic condition on its support. We will show that there exists a unique stationary probability measure ν on the projective space P(V ) corresponding to the top Lyapunov exponent. We describe the support of ν in terms of μ and relate it to the limit set of the semigroup of GL(V ) generated by the support of μ. Furthermore, we show that ν is Hölder regular and give some limit theorems of probabilistic flavor (as the exponential decay of the probability of hitting a hyperplane, exponential convergence in direction of the random walk). Our goal in the talk is to show how/why this setting
 is a natural one
 includes the wellknown “strongly irreducible and proximal” framework which was intensively studied and proved its efficiency, among others, in the study of the actions of semisimple algebraic groups
 gives new results when applied to a random walk on the affine group in the socalled “contracting case”
leads to new dynamics, essentially on a skewproduct space________________
Y.BENOIST and J.FQUINT: Random walk on linear groups
This course will focus on the asymptotic behavior of the product of random matrices chosen independently with the same law. We will explain methods used to prove limit laws for this product. In particular, we will enlight a few central results obtained by Y. Guivarc’h and his coauthors: the simplicity of the Lyapunov exponents, the multidimensional central limit theorem the Holder regularity of the Furstenberg stationary measure... _________________________
P. BOUGEROL: Random walks and representations of compact groups
A random walk on a group is given by products of independent random elements with the same distribution. It can provide some probabilistic intuition on the description of the irreducible representations of compact or complex groups. This occurs in an interpretation of Littelmann’s path model and leads to the DuistermaatHeckman measure through Brownian motion (work with Ph. Biane and N. O’Connell). On the other hand this description also follows from the tropicalization of a random walk on a semisimple group and geometric crystal theory (work of R. Chhaibi).______________________________________________________________
E. BREUILLARD: Spectral gaps and random walks
In this talk I will illustrate how the use of the probabilistic method and Guivarc’h ideas in the theory of random matrix products can be helpful in proving uniform spectral gap estimates for groups acting on finite or infinite homogeneous spaces of linear groups. Among other things it yields convenient proofs of key results of Varjú and SalehiVarjú on expanders, of BenoistSaxce on spectral gaps for compact groups, as well as of Poznansky’s C^{*}simplicity for linear groups. I will also discuss the uniformity of the spectral gap when both the measure and the (algebraic) action vary._____________________________________________________________________
C. BRUÈRE: Invariant measures on affine grassmannians
In joint work with Yves Benoist, we study the action of the affine group G of ^{d} on the affine Grassmannian X_{k,d}, that is, the set of affine kspaces in ^{d}. When G is endowed with a Zariskidense probability measure, we give a criterion for the existence of an invariant probability measure. Such a measure, if it exists, is unique.__________________________________________________________________________________________
R. CHHAIBI: Functional equations in number theory and the reflection principle for random walks
The general goal of this talk is to advertise the following curious point of view. I will argue that functional equations of Eisenstein series (or associated L functions) and the reflection principle for random walks are a manifestation of the same Weyl group symmetry. In the usual rank 1, I am saying that the two following symmetries are intimately related: the symmetry of the Riemann zeta function with respect to the critical axis and the symmetry used in order to count random walks conditioned to stay positive. In higher ranks (Eisenstein series associated to spherical representations of Lie groups), one can obtain such functional equations by proving that certain FourierWhittaker coefficients have a symmetry property. I will explain how this can be obtained from the reflection principle applied to suitable random walks on padic Lie groups. The general statement is that these FourierWhittaker coefficients are proportional to characters. The precise formula is known as the ShintaniCasselmanShalika formula for which we give a probabilistic proof (http://arxiv.org/abs/1409.4615). _________________________________________
A. ERSCHLER: Isoperimetry, diffusive random walks and quotients of groups
_______________________________________________________________________________________
A. GAMBURD: Random walks on MarkoffHurwitz surfaces
_______________________________________________________________________________________
E. KOWALSKI: The shapes of exponential sums
(Joint work with W. Sawin.) We will describe how the study of the behavior of certain exponential sums leads to questions about a very specific random Fourier series. In particular, we will describe some properties of the support of the limiting object, leading to a number of interesting questions and connections with classical Fourier analysis.___________________________________________________________________
P. MATHIEU: Deviation inequalities and CLT for random walks on hyperboliclike groups
(Joint work with A.Sisto.) We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be ”aligned”. We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. We also show that the (exponential) deviation inequality holds for measures with exponential tail on acylindrically hyperbolic groups. These include nonelementary (relatively) hyperbolic groups, Mapping Class Groups, many groups acting on CAT(0) spaces and small cancellation groups. ___________________________________________________________________
A. NEVO: New directions in Diophantine approximation
Classical Diophantine approximation quantifies the denseness of the set of rational vectors in their ambient Euclidean space. A farreaching extension of the classical theory would be to quantify the denseness of rational points (and rational points with constrained denominators) in general homogeneous algebraic varieties. This was raised as an open problem by Serge Lang already half a century ago, but progress towards it was achieved only in a limited number of special cases. A systematic approach to this problem for homogeneous varieties associated with simple groups has been developed in recent years, in joint work with A. Ghosh and A. Gorodnik, based on dynamical arguments and effective ergodic theory. This approach leads to the derivation of uniform and almost sure Diophantine exponents, as well as analogs of Khinchin’s and Schmidt’s theorems, with some of the results being best possible. We will explain some of the main results and some of the ingredients in their proof, focusing on some easily accessible examples._______________________________________________________________________________________
P. VARJÚ: Full dimension of Bernoulli convolutions
Fix a number 0 < λ < 1 and denote by ν_{λ} the stationary probability measure under the maps xλx + 1 and xλx  1. This measure is called the Bernoulli convolution with parameter λ. It is a long standing open problem to determine the set of parameters lambda for which ν_{λ} is absolutely continuous. I will discuss recent progress on this problem focusing on a joint work with Emmanuel Breuillard, which proves that if Lehmer’s conjecture holds, then there is a number a < 1 such that dim ν_{λ} = 1 for all λ in [a, 1). Unconditionally, we prove that dim ν_{λ} = 1 for some explicit examples of transcendental numbers λ such as ln(2) and π∕4.________________________________________________________________________________
Y. SHALOM: Uniform exponential escape of random matrix products from algebraic subgroups, and joining of graphs
We shall discuss a general nonconcentration result for random matrix products, which is a key ingredient in the vast literature on superstrong approximation. While this has always been a very technical step, our entirely different approach is much softer, and covers also, for the first time, the case of positive characteristics. Special role in the proof is played by a spectral extension of a classical theorem of Leighton: if two finite graphs admit the same universal covering (tree), then they admit a finite common covering. Based on joint work with Asaf Hadari.
__________________________________________________________________________
A. ZUK: Random walks on discrete KdV equations
Boxball systems are discrete analogues of the KdV equation. We prove that their evolution can be described by automata. With these automata we associate random walk operators. We relate spectral properties of these operators with L2 Betti numbers of closed manifolds.