Fall 2014
Monday 4.15  5.15 pm
Room E17139
Schedule

September 8
Paul Bourgade (Harvard)
Homogenization of the Dyson Brownian motion
Abstract: I will explain a homogenization result for the Dyson Brownian motion, which gives microscopic statistics from mesoscopic ones. It implies in particular the WignerDysonMehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed.

September 15
Samuel Watson (MIT)
A conformally invariant metric on CLE_{4}
Abstract: We will discuss an exploration process, introduced by Wendelin Werner and Hao Wu, in which conformal loop ensemble (CLE_{κ}) loops grow uniformly from the boundary of a domain. We relate this process in the case κ = 4 to the set of level loops of the zeroboundary Gaussian free field, and we use this point of view to show that the exploration process is a deterministic function of the CLE loops. We describe how this gives rise to a conformally invariant metric on CLE_{4}, which we conjecture can be given a natural geometric interpretation.
Joint work with Scott Sheffield and Hao Wu.

September 22
Allan Sly (UC Berkeley)
Increasing subsequences on the plane and the Slow Bond Conjecture
Abstract: For a Poisson process in R^{2} with intensity 1, the distribution of the maximum number of points on an oriented path from (0,0) to (N,N) has been studied in detail, culminating in BaikDeiftJohansson's celebrated TracyWidom fluctuation result. We consider a variant of the model where one adds, on the diagonal, additional points according to an independent one dimensional Poisson process with rate λ. The question of interest here is whether for all positive values of λ, this results in a change in the law of large numbers for the the number of points in the maximal path. A closely related question comes from a variant of Totally Asymmetric Exclusion Process, introduced by Janowsky and Lebowitz. Consider a TASEP in 1dimension, where the bond at the origin rings at a slower rate r<1. The question is whether for all values of r<1, the single slow bond produces a macroscopic change in the system. We provide affirmative answers to both questions.
Based on joint work with Riddhipratim Basu and Vladas Sidoravicius

October 6
Philippe Di Francesco (UIUC)
Whittaker functions and fusion product
Abstract: Whittaker functions have recently reappeared in the context of random polymers and special McDonald processes. We revisit their earlier life in representation theory: they were originally built out of Whittaker vectors, for which we give a new statistical weighted path formulation, valid for simple and affine Lie algebras as well as the quantum algebra U_{q}(sl_{n}). We show how this formulation bypasses the classical derivation of Todatype differential/difference equations satisfied by these functions. We then consider graded tensor products of current algebra modules, and show that their characters obey difference equations, generalizing the difference Toda equation, allowing for viewing graded characters as generalized Whittaker functions. This is done using a constant term expression for the characters via a solution of the quantum Qsystem, a set of noncommuting integrable recursion relations attached to the algebra. Finally, we obtain a new compact expression for graded sl_{n} characters by constructing a presentation of the quantum Qsystem via generalized McDonaldRuijsenaars difference operators. (based on joint works with R. Kedem, and R. Kedem and B. Turmunkh).

October 13
Columbus Day

October 17
Special full day event at Harvard!
The 2014 Charles River Lectures on Probability and Related Topics
The Charles River Lectures on Probability and Related Topics will be hosted by Harvard. The lectures are jointly organized by Harvard, MIT and Microsoft Research New England for the benefit of the greater Boston area mathematics community. The event features five lectures by distinguished researchers in the areas of probability and related topics.
This year's lectures will be delivered by:
David Brydges (University of British Columbia)
Sourav Chatterjee (Stanford University)
Christophe Garban (CNRS & ENS Lyon, UMPA)
Fabio Toninelli (CNRS & Institut Camille Jordan, Lyon 1)
Srinivasa Varadhan (New York University) 
October 20
Maurice Duits (Stockholm University)
CLT's for global and local linear statistics in orthogonal polynomial ensembles
Abstract: In this talk, I will present some recent joint work with Jonathan Breuer on the fluctuations of linear statistics in orthogonal polynomial ensembles. Such ensembles appear naturally in random matrix theory and integrable probability and the fluctuations of linear statistics have been studied in various contexts. I will discuss a new approach for studying such fluctuations based on the recurrence coefficients or Jacobi operator corresponding to the orthogonal polynomials. The main results are Central Limit Theorems for linear statistics on both the global and local scales (or macro and mesocopic scales) under rather mild conditions on the underlying measure. If time permits, I will also discuss the extension of the CLT on the global scale to biorthogonal ensembles.

October 27
Ofer Zeitouni (Weizmann Institute)
On roots of random polynomials
Abstract: I will discuss some recent results concerning the distribution of roots of random polynomials, focusing on results concerning roots of systems of polynomials. In particular, I will discuss the probability that a random Littlewood polynomial has multiple roots. If time permits, I will also discuss a nonstandard large deviation result for random polynomials with positive real coefficients.
(Based on joint works with G. Kozma, with R. Peled and A. Sen, and with S. Ghosh)

November 3
Madhu Sudan (Microsoft Research New England)
Communication with Imperfectly Shared Randomness
Abstract: In the 1980's Yao introduced the model of communication complexity where Alice, who knows x in {0,1}^{n}, and Bob, who knows y in {0,1}^{n}, wish to communicate with each other, exchanging fewest possible number of bits, to determine some function f(x,y). For many natural functions f this communication complexity is much smaller than the trivial n bits suggesting that communication with a goal in mind may be much less expensive than otherwise. This message gets amplified even more if Alice and Bob share some random string r chosen independently of x and y: in this setting even more functions f can be determined quickly (with high probability).
In this talk I will introduce a relaxation of this model where Alice and Bob do not share the random string perfectly, but rather Alice knows r and Bob knows some string s that is correlated with r. I will describe some recent results showing that any communication protocol with k bits of communication between Alice and Bob with perfect sharing of randomness, continues to have a moderately lowcomplexity, 2^{k} bit, protocol with shared correlation. Furthermore, I will show that this result is tight in that there exist problems where this exponential jump is necessary. The technical core of these results rely on the understanding of the influence of variables in the analysis of Boolean functions and recently developed probabilistic tools such as the "invariance principle" of Mossel, O'Donnell and Oleszkiewicz. One hope of the talk is to explain what these tools are, and why computer scientists find them useful.
Based on joint work with Clément Canonne (Columbia), Venkatesan Guruswami (CMU) and Raghu Meka.

November 13
Thursday,Room E18466A, at 2:30
Yuval Peres (Microsoft Research)
Rigidity and tolerance for perturbed lattices
Abstract: Consider a perturbed lattice {v+Y_{v}} obtained by adding IID ddimensional Gaussian variables {Y_{v}} to the lattice points in Z^{d}. Suppose that one point, say Y_{0}, is removed from this perturbed lattice; is it possible for an observer, who sees just the remaining points, to detect that a point is missing? In one and two dimensions, the answer is positive: the two point processes (before and after Y_{0} is removed) can be distinguished using smooth statistics, analogously to work of Sodin and Tsirelson (2004) on zeros of Gaussian analytic functions. (cf. Holroyd and Soo (2011) ). Further rigidity results for these zeros were obtained with Ghosh (2012), based on an idea of Nazarov and Sodin. The situation in higher dimensions is more delicate; our solution depends on a gametheoretic idea, in one direction, and on the unpredictable paths constructed by Benjamini, Pemantle and the speaker (1998), in the other. I will also describe a related point process where removal of one point can be detected but not the removal of (any!) two points.
(Joint work with Allan Sly, UC Berkeley).

November 17
Russell Lyons (Indiana University)
Random walks on groups and the KaimanovichVershik conjecture
Abstract: Let G be an infinite group with a finite symmetric generating set S. The corresponding Cayley graph on G has an edge between x,y in G if y is in xS. KaimanovichVershik (1983), building on fundamental results of Furstenberg, Derriennic and Avez, showed that G admits nonconstant bounded harmonic functions iff the entropy of simple random walk on G grows linearly in time; Varopoulos (1985) showed that this is equivalent to the random walk escaping with a positive asymptotic speed. Kaimanovich and Vershik also presented the lamplighter groups (groups of exponential growth consisting of finite lattice configurations) where (in dimension at least 3) the simple random walk has positive speed, yet the probability of returning to the starting point does not decay exponentially. They conjectured a complete description of the bounded harmonic functions on these groups; in dimensions 5 and above, their conjecture was proved by Erschler (2011). I will discuss the background and present a simple proof of the KaimanovichVershik conjecture for all dimensions, obtained in joint work with Yuval Peres.

November 20
Thursday, Room E17133 at 3:00
Adrian Banner (INTECH)
Rankbased portfolios, the "size effect", and an identity for the exponential distribution
Abstract: In an equity market with stable capital distribution, a capitalizationweighted index of small stocks tends to outperform a capitalizationweighted index of large stocks. This is a somewhat careful statement of the socalled “size effect”, which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder.
We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the socalled “Atlas model”. As a corollary we shall obtain a hithertounknown identity for the exponential distribution. (Joint work with R. Fernholz, I. Karatzas, V. Papathanakos and P. Whitman.)

November 24
at 3:40
Natesh Pillai (Harvard)
Probabilistic Challenges in MCMC algorithms
Abstract: (Markov Chain Monte Carlo) algorithms are an extremely powerful set of tools for sampling from complex probability distributions. In this two talk we will discuss two research themes for studying the efficiency of commonly used algorithms. We will discuss optimal scaling of MCMC algorithms in high dimensions where the key idea is to study the properties of the proposal distribution as a function of the dimension. This point of view gives us new insights on the behavior of the algorithm, such as precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. In the second part, we will introduce `adaptive Markov chains', and discuss their finite sample properties.

December 1
Leonid Koralov (UMD)
Transition between averaging and homogenization regimes for periodic flows and averaging for flows with ergodic components
Abstract: In this talk we'll discuss two asymptotic problems that are related by common techniques. First, we'll consider elliptic PDEs with with a small diffusion term in a large domain. The coefficients are assumed to be periodic. 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. The second problem concerns equations with a small diffusion term, where the firstorder term corresponds to an incompressible flow, possibly with a complicated structure of flow lines. Here we prove an extension of the classical averaging principle of Freidlin and Wentzell.
Different parts of the talk are based on joint results with M. Hairer, Z. PajorGuylai, D. Dolgopyat, and M. Freidlin.

December 8
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 byproduct, 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 excursionreflected Brownian motions, which have arisen in the study of SLE. This talk is based on works with Chris Burdzy, Zhenqing Chen and Donald Marshall.
Semester/Year Programs

January 5April 3, 2015
IHP trimester on Disordered Systems, Random Spatial Processes and their Applications, Paris, France
This trimester will include the following workshops:
26 January 2015  30 January 2015
Statistical physics methods in social and economic systems16 February 2015  20 February 2015
Spin glasses, random graphs and percolation09 March 2015  13 March 2015
Interacting particles systems and nonequilibrium dynamics 
February 2May 8, 2015
ICERM Semester Program on Phase Transitions and Emergent Properties at Brown University
Schools

January 59, 2015
Introductory school at CIRM (Marseille, France) : Disordered systems, random spatial processes and some applications
Conferences and Lecture Series

August 611, 2014
7th International Conference on Stochastic Analysis and its Applications, Seoul, Korea

August 1321, 2014
International Congress of Mathematicians, Seoul, Korea

August 2226, 2014
International Conference on Quantum Probability and Related Topics, Seoul, Korea
Fall 2014 Organizers
 Alexei Borodin
 Vadim Gorin
 Alice Guionnet
 Jason Miller
 Scott Sheffield
 Nike Sun
 Omer Tamuz
 Hao Wu