Alexandre Stauffer (Berkeley)

Motivated by mobile wireless networks we consider a random graph over R^2, where nodes perform independent Brownian motions and edges are kept for pairs of nodes within distance r of each other. Combining ideas from stochastic geometry, coupling and multi-scale analysis, we obtain precise asymptotics for detection (the time until a given target is within distance r to some node of the graph), coverage (the time until all points inside a finite box are detected), and percolation (the time until a given node belongs to the infinite connected component of the graph). (This is a joint work with Yuval Peres, Alistair Sinclair and Perla Sousi.)
top of page

Gerard Ben Arous (Courant Institute, NYU)

I will present a joint work with Paul Bourgade (Harvard) about the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices (CUE) and matrices from the Gaussian Unitary Ensemble. In particular, we show that the smallest gaps when rescaled by N^-4/3, are Poissonian and we give the limiting distribution of the kth smallest gaps. We also show that the largest gap, when normalized by ?log N/N, converges in L^p to a constant for all p > 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.
top of page

James R. Lee (U. of Washington)

The cover time of a graph is one of the most basic and well-studied properties of the simple random walk, and yet a number of fundamental questions concerning cover times have remained open. We show that there is a significant connection between cover times of graphs, discrete Gaussian free fields, and Talagrand's theory of majorizing measures. In particular, we prove that the cover time can be characterized, up to universal constants, by the majorizing measure value of the resistance metric on the underlying graph.
We use this connection to resolve the blanket time conjecture of Winkler and Zuckerman (1996) and a question of Aldous and Fill (1994) on whether there is an efficient deterministic algorithm that approximates the cover time of any graph.
(Joint work with Jian Ding and Yuval Peres.)
top of page

Ofer Zeitouni (U. of Minnesota)

The Gaussian free field (GFF) on a finite graph plays an important role in many aspects of contemporary probability theory as well as in mathematical physics, through models for random interfaces and quantum gravity.
Branching random walks (BRW's) model the evolution of a population, where particles split and then perform independent random motion in R. As shown by Bramson in the 70's, the law of the particle that moved farthest from the starting point is determined, in the Gaussian displacement case, by solutions of the Kolmogorov-Petrovsky-Piscounov PDE. In the non-Gaussian case, a proof that fluctuations of the maximum are of O(1) was given only very recently.
I will describe surprising links that BRW's have with GFF's, first passage percolation, and the cover time of graphs by random walks. I will then explain how arguments developed for the study of BRW's play a role in a recent resolution of the conjecture that fluctuations of the maximum of the two-dimensional GFF are bounded.
Based on joint works with Bolthausen and Deuschel, and with Bramson.
top of page

Ivan Corwin (Courant Institute, NYU)

The Kardar-Parisi-Zhang (KPZ) stochastic PDE is the central object in the study of random growth models, interacting particle systems and random polymer models. Bertini and Giacomin provide an approach for defining and approximating this SPDE (properly interpreted) as the limit of an interacting particle system -- the weakly asymmetric exclusion process. Tracy and Widom provide certain exact transition formulas for this particle system. Combining these two approaches we are able to derive and prove the first exact formula for the distribution function for the solution to the KPZ equation.
This talk is based on joint work with Gideon Amir and Jeremy Quastel.
top of page

Alice Guionnet (Ecole Normale Supérieure de Lyon)

I will show how to construct matrix models related with the generating functions for loop models and how to use this construction to indeed compute these generating functions in special cases such as the Potts model.
top of page

Jason Miller (Stanford)

The discrete Gaussian free field (DGFF) is the Gaussian measure on real-valued functions h(.) on a bounded subset D of the two dimensional integer lattice, whose covariance is given by the Green's function for simple random walk. The graph of h(.) is a random surface which serves as a physical model for an effective interface. We show that the collection of random loops given by the level sets of the DGFF at any height converges in the fine-mesh scaling limit to a family of loops which is invariant under conformal transformations when D is a lattice approximation of a non-trivial simply connected domain. In particular, there exists lambda>0 such that the level sets whose height is an odd integer multiple of lambda converges to a nested conformal loop ensemble with parameter kappa=4 [so-called CLE(4)], a conformally invariant measure on loops which locally look like SLE(4). Using this result, we give a coupling of the continuum Gaussian free field (GFF), the fine-mesh scaling limit of the DGFF, and CLE(4) such that the GFF can be realized as a functional of CLE(4) and conversely CLE(4) can be made sense as a functional of the GFF.
top of page

Isabelle Camilier (Stanford)

Determinantal processes are point processes with a correlation function given by a determinant. Their atoms exhibit mutual repulsion. We establish a quasi-invariance result: we show that if atom locations are perturbed along a vector field, the resulting process is still a determinantal process, the law of which is absolutely continuous with respect to the original distribution. Then, based on this, we give an integration by parts formula.
top of page

Jose Blanchet (Columbia)

We consider two related problems: a) efficient Monte Carlo evaluation of the tail of the maximum of a Gaussian process and b) sampling the process conditional on observing a high excursion. In great generality, the tail of such maximum decreases exponentially fast in a power of the underlying tail parameter, so that naive Monte Carlo takes an exponential amount of time to run. We show algorithms that possess strong optimality properties (i.e. polynomial running time in the underlying tail parameter) under mild assumptions. For instance, we start by discussing the maximum of an infinite horizon Gaussian process in discrete time and show optimality of our algorithms under an arbitrary correlation structure. Then, we discuss questions a) and b) in the setting of Gaussian random fields that are only assumed to be Holder continuous. Our methodology relies on importance sampling but, contrary to standard uses of this technique in the Gaussian setting, we do not rely on an exponential changes of measure. (This talk is based on joint work with R. Adler, J. C. Liu and C. Li).
top of page

Emmanuel Candes (Stanford)

In the last few years, we have seen the development of an exciting field, which addresses a broad range of problems of great practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. This is of interest because this problem occurs in many areas of engineering and applied science such as machine learning, control, statistics, data mining, and computer vision. A concrete example may be the problem of inferring the many missing entries in a partially filled out survey. Some recent results assert that it is possible to recover low-rank matrices exactly from what appear to be highly incomplete sets of sampled entries by simply minimizing the nuclear norm. Others assert that exact recovery via convex programming is further possible even in situations where a positive fraction of the observed entries are corrupted in an almost arbitrary fashion.
All these results are probabilistic in nature, and the goal of this technical talk is to give some insights about why they are true and sketch some proofs. Our goal is to present the main mathematical ideas with an emphasis on powerful tools from probability theory.
top of page