Schedule

• February 9 February 17

  Change of date and venue due to storm on the 7^th. Room E17-122.

  Philippe Sosoe (Harvard)

  On the chemical distance in critical percolation

  Abstract: In two-dimensional critical percolation, the works of Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+ε, for some positive ε. No more precise lower bound has been given so far. Conditional on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Zhang and Morrow to have volume n^4/3+o(1) on the triangular lattice.

  Following a question of Kesten and Zhang, we compare the length of shortest circuit in an annulus to that of the innermost circuit (defined analogously to the lowest crossing). I will explain how to show that the ratio of the expected length of the shortest circuit to the expected length of the innermost crossing tends to zero as the size of the annulus grows.

  Joint work with Jack Hanson and Michael Damron.

• February 23

  Alexander Holroyd (MSR)

  Finitely dependent coloring

  Abstract: A central concept of probability and ergodic theory is mixing in its various forms. The strongest and simplest mixing condition is finite dependence, which states that variables at sufficiently well separated locations are independent. A 50-year old conundrum is to understand the relationship between finitely dependent processes and block factors (a block factor is a finite-range function of an independent family). The issue takes a very surprising new turn if we in addition impose a local constraint (such as proper coloring) on the process. In particular, this has led to the discovery of a beautiful yet mysterious stochastic process that seemingly has no right to exist.

• March 2

  Nick Gravin (MSR)

  Towards Optimal Algorithms for Prediction with Expert Advice

  Abstract: We study the classical problem of prediction with expert advice in the adversarial setting with a geometric stopping time. In 1965, Cover gave the optimal algorithm for the case of 2 experts. In this work, we design the optimal algorithm, adversary and regret for the case of 3 experts. Further, we show that the optimal algorithm for 2 and 3 experts is a probability matching algorithm (analogous to Thompson sampling) against a particular randomized adversary. It turns out that this algorithm is not only optimal against this adversary, but also minimax optimal against all possible adversaries.

  We establish a constant factor separation between the regrets achieved by the optimal algorithm and the widely used multiplicative weights algorithm. Along the way, we improve the regret lower bounds for the multiplicative weights algorithm for an arbitrary number of experts and show that this is tight for 2 experts. In our analysis, we develop upper and lower bounds simultaneously, analogous to the primal-dual method. The analysis of the optimal adversary relies on delicate random walk estimates. We further use this connection to develop an improved regret bound for the case of 4 experts, and provide a general framework for designing the optimal algorithm for an arbitrary number of experts.

  This is a joint work with: Yuval Peres, Balasubramanian Sivan

• March 9

  Senya Shlosman (Marseille)

  Ising random walks and their pinning properties

  Abstract: Ising random walk is a random walk with certain self-interaction. It describes the behavior of the interface, separating the (+)-phase from the (-)-phase of the 2D Ising model. One says that pinning is taking place, when, due to the self-interaction, the qualitative behavior of the random walk is different from the non-interacting case. I will explain when pinning does happen and when it does not happen.

  Based on a joint work with D. Ioffe and F. Toninelli, arXiv:1407.3592

• March 23

  Spring vacation

• March 30

  Boris Hanin (MIT)

  Pairing Between Zeros and Critical Point of Random Polynomials

  Abstract: Let p_N be a degree N polynomial in one complex variable. The purpose of this talk is to explain why the zeros and critical points of p_N come in pairs, spaced about N^-1 apart. I will explain how such a pairing arises by relating zeros and critical points to electrostatics on the Riemann sphere.

• April 6

  Ashkan Nikeghbali (University of Zurich)

  Ratios in random matrix theory and number theory

  Abstract: We shall consider the Circular Unitary Ensemble and ratios of characteristic polynomials at the microscopic scale in this ensemble. These ratios have been extensively studied in the past decade, using various techniques and approaches, in problems related for instance to mathematical physics or number theory (value distribution of the Riemann zeta function). It has been observed by various authors that the average of these ratios and their logarithmic derivatives converge (at the microscopic scale) when the size of the unitary group goes to infinity. In this talks we shall provide a probabilistic framework from which we deduce that the ratios themselves converge to some explicit random rational function and we compute explicitly the average of this object. We also explain how we expect these limiting objects to be related to corresponding ratios for the Riemann zeta function.

• April 9

  Thursday, 3 to 4pm, room 66-168

  Dmitry Ioffe (Technion)

  A quantitative Burton-Keane estimate under strong FKG condition

  Abstract: Abstract: We consider translation-invariant percolation models on Z^d satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance n (this corresponds to a finite size version of the celebrated Burton-Keane argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincare inequality for Bernoulli percolation which was derived by Chatterjee and Sen. As a consequence, we show how RSW-type estimates recently obtained by Duminil-Copin, Sidoravicius and Tassion imply upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight q in [1,4].

  Joint work with Hugo Duminil-Copin and Yvan Velenik

• April 27

  Kavita Ramanan (Brown University)

  Obliquely reflected diffusions in rough planar domains

  Abstract: Obliquely reflected diffusions in smooth domains are classical objects that have been well understood for half a century. Motivated by applications in a variety of fields ranging from mathematical physics to stochastic networks, a theory for obliquely reflected diffusions in piecewise smooth domains has also been developed over the last two decades. However, in domains with rough boundaries, even the definition of obliquely reflected diffusions is a challenge. We discuss an approach to constructing obliquely reflected Brownian motions (ORBMs) in a large class of bounded, simply connected planar domains that, as a by-product, also provides a new characterization of ORBMs in bounded smooth planar domains. The class of processes we construct also includes certain processes with jumps like excursion-reflected Brownian motions, which have arisen in the study of SLE. This talk is based on works with Chris Burdzy, Zhenqing Chen and Donald Marshall.

• May 4

  3 to 4pm

  Robin Pemantle (University of Pennsylvania)

  Three problems in discrete probability

  Abstract: I will talk for roughly 5 minutes on each of three topics, after which the audience will decide which topic should occupy the remaining half+ hour.

  Topic #1: start with a Poisson process of particles on the line; evolve by a rule in which large intervals eat smaller intervals; what happens?

  Topic #2: Make a random set M containing each positive integer n independent with probability 1/n. Let S be the set of sums of elements of M. How many independent copies of S must be intersected to arrive at a finite set, and what does this question have to do with computational Galois theory?

  Topic #3: Construct a random entire function, f, whose zeros form a unit intensity Poisson process on the real line. Differentiate f repeatedly, and let L_n be the joint law of the zeros of the nth derivative of f. What is the limit as n → infinity of L_n, and why?

• May 11

  Ioana Dumitriu (University of Washington)

  Two clustering problems involving the Stochastic Block Model

  Abstract: Clustering and community detection in large networks are important problem with a wide spectrum of applications, from marketing to machine learning and from genomics to social sciences. Clustering, the non-overlapping case, has been studied for decades in settings increasingly general; one of the most widely-used models is the Stochastic Block Model (SBM), where the set of vertices is partitioned in a number of subsets, and an independent Erdos-Renyi graph is constructed on each of these sets; the inter-set edges are then given by an independent multipartite Erdos-Renyi graph. Given the adjacency matrix of the large graph, the problem consists of devising an algorithm that identifies (correctly or approximately) the original vertex partition (or showing no such algorithm exists).

  Even when making several simplifications, the problem is difficult enough that the only case that has been completely solved (in the sense that all parameter reginmes have been identified) is the two equal-sized set case (binary, balanced SBM) through a concerted effort by Mossel, Neeman and Sly, parallelled by Massoulie. Inspired by their work, we have considered and analyzed a regular binary SBM, where the Erdos-Renyi models are replaced by uniform regular graphs. The first part of my talk will focus on this work, joint with Gerandy Brito, Shirshendu Ganguly, Christopher Hoffman, and Linh Tran.

  The second part of my talk will involve recent work on a general SBM model involving recovery regimes (sets of parameters for which correct identification of the partition is possible). This work is joint with Maryam Fazel, Roy Han, and Amin Jalali.

• May 18

  Claudio Landim (IMPA)

  Macroscopic fluctuation theory

  Abstract: Non-equilibrium stationary states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems.

Semester/Year programs

• February 2-May 8, 2015

  ICERM Semester Program on Phase Transitions and Emergent Properties at Brown University

Schools

• January 5-9, 2015

  Introductory school at CIRM (Marseille, France) : Disordered systems, random spatial processes and some applications

Conferences and Lecture Series

• August 6-11, 2014

  7th International Conference on Stochastic Analysis and its Applications, Seoul, Korea

• August 13-21, 2014

  International Congress of Mathematicians, Seoul, Korea

• August 22-26, 2014

  International Conference on Quantum Probability and Related Topics, Seoul, Korea