# Abstracts and Slides

D. AHLBERG     Quenched Voronoi percolation
Slides
In a seminal work from 1999, Benjamini, Kalai and Schramm introduced a
framework for studying sensitivity of Boolean functions with respect
to small portions of noise. They moreover made a series of conjectures
that have been highly influential for the development since. We will discuss
this development in some detail, and look more closely at the recent
solution to one of these conjectures, concerning Voronoi percolation:
Position a large number of points in the unit square and consider their
Voronoi tessellation. Next, colour each cell either red or blue. The question
is whether observing the tessellation, but not the colouring, will help us in
guessing whether the colouring will produce a horizontal red crossing or
not? We establish that this is not the case.

E. AÏDÉKON     Scaling limit of the recurrent biased random walk on a Galton-Watson tree
We consider a biased random walk on a Galton-Watson tree. This Markov chain is null recurrent for a critical value of the bias. In that case, Peres and Zeitouni proved that the height of the Markov chain properly rescaled converges in law to a reflected Brownian motion $B$. We show here that the trace of this Markov chain converges in law to the Brownian forest encoded by $B$. Joint work with Loïc de Raphélis.

L.-P. ARGUIN     Fluctuation bounds for interface free energies in spin glasses
Slides
One way to understand the structure of the Gibbs states of disordered systems is to get good bounds on the fluctuations of the free energy difference between two states. This approach has led to the proof of the absence of phase transition in the 2D Random Field Ising Model (RFIM) at the end of the 1980’s. We will explain a method to obtain lower bounds for the variance of the free energy difference of the Edwards-Anderson (EA) spin glass model on Z^d between certain incongruent states (if they exist…). Unlike the RFIM, there is no dominance of the (+) and (-) states in the EA model. One interesting point of the method is to overcome this lack of monotonicity. The lower bound is also used to rule out particular structures of the Gibbs states in d=2. This is joint work with D. Stein, C. Newman and J. Wehr.

A. AUFFINGER     Recent results on mean field spin glass model
I will survey recent and ongoing progress on mean field spin glasses. I will primarily discuss the behavior and role of the Parisi measure in the SK model, and its connections to ultrametricity and chaos.
Based on joint works with Wei-Kuo Chen (U. Minnesota).

Y. BAKHTIN     Ergodic theory of Burgers equation with random forcing
Slides

I will talk about extending the ergodic theory of randomly forced Burgers equation (a basic nonlinear evolution PDE related to hydrodynamics and growth models) to the noncompact setting. In the inviscid case, a variational principle holds, so an essential part of the program is constructing one-sided infinite minimizers of random action and studying their properties. The corresponding results are joint with Eric Cator and Kostya Khanin for Poissonian forcing and due to myself in the kicked forcing case. I will also report on the progress for the viscous case (joint with Liying Li). Here the variational characterization is replaced by the Feynman-Kac formula, so a natural approach is to construct and study infinite-volume polymer measures.

P. BOURGADE      The eigenvector moment flow and applications
Slides
For generalized Wigner matrices, I will explain a probabilistic version of quantum unique ergodicity at any scale, and gaussianity of the eigenvectors entries. The proof relies on analyzing the effect of the Dyson Brownian motion on eigenstates.  Relaxation to equilibrium of the eigenvectors is related to a new multi-particle random walk in a random environment, the eigenvector moment flow. Applications of this local quantum unique ergodicity to universality will be mentioned. This is joint work with H.-T. Yau.

A. BOVIER    Extremes of Gaussian Processes on Trees
Slides
Gaussian processes indexed by trees form an interesting class of correlated random fields where the structure of extremal processes can be studied.
One popular example is Branching Brownian motion, which has received a lot of attention over the last decades, non the least because of its connection to the KPP equation.
In this talk I review the construction of the extremal process of standard and variable speed BBM (with Arguin, Hartung, and Kistler).

N. CURIEN     First passage percolation on random planar maps
Slides
Random planar maps are a model of random two dimensional discrete geometry studied a lot in recent years and motivated by two dimensional quantum gravity.  We prove a general principle showing that perturbing locally the graph distance in a random planar map does not change the large scale geometric structure up to multiplying it by a deterministic constant. This applies in particular to perturbations of the metric obtained by first-passage percolation or by taking the dual of the map. The dilation constant is given by a subtle subadditive lemma taking place in a half-planar version of these maps.  In certain cases such as first-passage percolation with exponential edge weights on the dual (known as Eden model in the literature), or Tutte’s classical bijection between quadrangulations and general maps we are even able to compute explicitly the dilation constant. Based on joint works with J.F. Le Gall.

P. DEY     Longest increasing path within the critical strip
Consider a Poisson Point Process of intensity one in the two-dimensional square $[0,n]^2$. In Baik-Deift-Johansson (1999), it was shown that the length $L_n$ of a longest increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later Johansson (2000) showed that all maximal paths lie within the strip of width $n^{2/3+\epsilon}$ around the diagonal with probability tending to $1$ as $n\to \infty$. We consider the length $L_n^{(\gamma)}$ of maximal increasing paths restricted to lie within a strip of width $n^{\gamma}, \gamma<2/3$ around the diagonal and show that when properly centered and scaled it converges to a Gaussian distribution. We also obtain tight bounds on the expectation and variance of $L_n^{(\gamma)}$. Joint work with Matthew Joseph and Ron Peled.

P. FERRARI     Phase transition for the diluted clock model
Slides
We prove that phase transition occurs in the dilute ferromagnetic nearest-neighbour q-state clock model in Z^d, for every q≥2 and d≥2. This follows from the fact that the Edwards-Sokal random-cluster representation of the clock model stochastically dominates a supercritical Bernoulli bond percolation probability, a technique that has been applied to show phase transition for the low-temperature Potts model. The domination involves a combinatorial lemma which has interest by itself. Joint work with Inés Armendariz and Nahuel Soprano-Loto.

M. GUBINELLI     Regularisation by noise in some PDEs
Slides

We discuss some examples of the “good” effects of “very bad”, “irregular” functions. In particular we will look at non-linear differential (partial or ordinary) equations perturbed by noise. By defining a suitable notion of “irregular” noise we are able to show, in a quantitative way, that the more the noise is irregular the more the properties of the equation are better. Some examples includes: ODE perturbed by additive noise, linear stochastic transport equations and non-linear modulated dispersive PDEs. It is possible to show that the sample paths of Brownian motion or fractional Brownian motion and related processes have almost surely this kind of irregularity. (Joint work with R. Catellier and K. Chouk).

L. KORALOV     Averaging, homogenization, and large deviation results for the study of randomly perturbed dynamical systems
Slides
In this talk we’ll discuss several asymptotic problems that can be formulated in terms of PDEs and solved using probabilistic methods. The first set of problems concerns the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term. Here we employ an extension of the large-deviation theory of Freidlin and Wentzell. Another set of problems concerns equations with a small diffusion term, where the first-order term corresponds to an incompressible flow, possibly with a complicated structure of flow lines. Here we use an extension of the averaging principle. Finally, we’ll consider equations with a small diffusion term with periodic coefficients in a large domain. Depending on the relation between the parameters, either averaging or homogenization need to be applied in order to describe the behavior of solutions. We’ll discuss the transition regime. Different parts of the talk are based on joint results with D. Dolgopyat, M. Freidlin, M. Hairer, and Z. Pajor-Guylai.

B. RIDER     Spiking the random matrix hard edge
Slides
The largest eigenvalues of a finite rank perturbation of a random hermitian matrix are known to exhibit a phase transition. If the perturbation is “small” one sees Tracy-Widom behavior, while a “large” perturbation results in Gaussian effects (with a scaling window about the critical value leading to a separate interpolating family of limit laws). This basic discovery is due to Baik, Ben Arous, and Peche at “beta=2”, with Bloemendal and Virag later showing the picture persists in the context of the general beta ensembles. Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

L. ROLLA     Absorbing-State Phase Transitions
Modern statistical mechanics offers a large class of driven-dissipative stochastic systems that naturally evolve to a critical state, of which activated random walks is perhaps the best example. The main pursuit in this field is to describe the critical behavior, the scaling relations and critical exponents of such systems, and whether the critical density in the infinite system is the same as the equilibrium density in their driven-dissipative finite-volume version. These questions are however far beyond the reach of existing techniques. In this talk we will report on the progress obtained in recent years, and discuss some of the open problems.

A. SEN     Double Roots of Random Littlewood Polynomials
Slides
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions.
This is joint work with Ohad Feldheim, Ron Peled and Ofer Zeitouni.

A. STAUFFER     Random walk on dynamical percolation
Slides

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed, and refresh their status at rate \mu. At the same time a random walker moves on G at rate 1 but only along edges which are open. The regime of interest here is when \mu goes to zero as the number of vertices of G goes to infinity, since this creates long-range dependencies on the model. When G is the d-dimensional torus of side length n, we prove that in the subcritical regime,  the mixing times is of order n^2/\mu. We also obtain results concerning mean squared displacement and hitting times. This is a joint work with Yuval Peres and Jeff Steif.

L. TOURNIER     Activated random walks with bias
Slides

The Activated Random Walk model is a conservative particle system in which particles move independently except that, when alone at a vertex, particles may switch to a passive state and then stay still until the visit of another particle. The competition between local deactivation and global spread of activity by diffusion is believed to lead, in wide generality, to a nontrivial phase transition as the initial density increases: at low density, local configurations eventually stabilize, while at higher density activity persists locally forever. The case when the particle motion is biased was first specifically considered by Taggi. In this talk, we will present an extension of this result. This is joint work with Leonardo Rolla.

M. E. VARES      Layered systems at the mean field critical temperature
Slides
In this talk, I will report on a recent work in collaboration with L.R. Fontes, D. Marchetti, I. Merola, and E. Presutti. We consider the Ising model on $\mathbb Z\times \mathbb Z$ where on each horizontal line $\{(x,i), x\in \mathbb Z\}$, the interaction is given by a ferromagnetic Kac potential with coupling strength $J_\ga(x,y)\sim\gamma J(\gamma (x-y))$ at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $\epsilon$ and prove that for every $\epsilon >0$ the systems exhibits phase transition provided $\gamma>0$ is small enough.

J. YIN     Comparison method in random matrix theory
Slides

Comparison method has been used in the proofs of many theorems in random matrix theory, e.g., bulk universality, edge universality of Wigner matrix, local circular law. In previous work, most comparison work is based on Lindeberg strategy. In this talk, we will introduce a new comparison method: continuous self consistent comparison method, and discuss its applications on random matrix theory.

O. ZEITOUNI     Large deviations, random matrices and random surfaces
Slides
I will describe recent progress in the study of random Gaussian functions in high dimensions. Special emphasis will be on the study of the extremal process of maxima. A key tool is the study of  certain determinants involving random matrices.
Based on work of E. Subag and on ongoing work with him and with G. Ben Arous.