Schedule

• January 30

  Horng-Tzer Yau (Harvard University)

  Random Matrix, Beta ensembles, and Dyson Brownian Motion

  Abstract: The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large random matrices in the bulk exhibituniversal behavior depending only on the symmetryclass of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log gas with a potential $V$ and inverse temperature $\beta = 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent of $V$ for all positive real $\beta$.In this talk, we review our recent solution to the universality conjecture for both invariant and non-invariant ensembles. The special role played by the logarithmic Sobolev inequalityand Dyson Brownian motion will be discussed.

• February 6

  Mirjana Vuletic (Brown University)

  The Gaussian free field and Pfaffian processes

  Abstract: The Gaussian free field is a random field that is associated with many random surface models. The goal of this talk is to explain how they arise from Pfaffian point processes. I will introduce a measure on plane partitions that is a Pfaffian point process. I will show that the height fluctuations around the limit shape converge to a pullback of the Gaussian process whose covariance is given by the Green's function for the Laplacian with Dirichlet boundary conditions on the first quadrant. The result was obtained by computing higher moments using the steepest descent analysis, based on an idea of Kenyon. The argument can be generalized to a class of Pfaffian processes whose kernels possess certain properties.

• February 13

  Davar Khoshnevisan (University of Utah)

  On the chaotic character of some parabolic SPDEs

  Abstract: It has been observed by several authors that a large family of stochastic partial differential equations have solutions that are highly intermittent; this means that the solution tends to develop tall peaks that are distributed oversmall sets ["islands"]. In this talk we will argue that the solution of such stochastic PDEs can be "chaotic," and that thisproperty leads to the onset of intermittence.

  This talk is based on joint works with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu.

• February 20

  No Seminar (Presidents day)

• February 22

  (SPECIAL DAY 4.15pm in 2-142)

  Benedek Valko (University of Wisconsin, Madison)

  Point processes and carousels

  Abstract: For several classical matrix models the joint density of the eigenvalues can be written as an expression involving a Vandermonde determinant raised to the power of 1, 2 or 4. Most of these examples have beta-generalizations where this exponent is replaced by a parameter beta>0. In recent years the point process limits of various beta ensembles have been derived. The limiting processes are usually described as the spectrum of certain stochastic operators or with the help of a coupled system of SDEs.In the bulk beta Hermite case (which is the generalization of GUE) there is a nice geometric construction of the point process involving a Brownian motion in the hyperbolic plane, this is the Brownian carousel. Surprisingly, there are a number of other limit processes that have carousel like representation. We will discuss a couple of examples and some applications of these new representations.

  Joint with Balint Virag.

• February 27

  Robin Pemantle (University of Pennsylviania)

  Zeros of complex polynomials and their derivatives

• March 5

  Hilario (UFMG Belo Horizonte, Brazil)

  Cylinders' Percolation in three dimensions

  Abstract: We consider the model of cylinders' percolation in three dimensions. This model was introduced by I. Benjamini and consists of removing from R^d a set of infinite cylinders having radius one and axis given by the lines in the support of a Poisson point process in the space of lines of R^d. An nonnegative intensity parameter u controls the amount of cylinders to be removed. In 2010, J. Tykesson and D. Windisch showed that for d>3 the connectivity of the complementary set undergoes a phase transition as u varies. They also showed that for d=3 and u large enough there is no percolation.In this work we prove the existence of a percolative phase for d=3. More than that, we prove that the complementary set percolates in a sufficiently thick slab if u is small enough, in spite of the fact that its restriction to any 2-dimensional subspace does not percolate regardless of the value of u.

  Joint work with A. Teixeira (ENS and IMPA) and V. Sidoravicius (IMPA).

• March 9

  SPECIAL SEMINAR: Friday from 2pm-3pm in room 2-143

  Daniel Kane (Stanford)

  Diffuse Decompositions of Polynomials

  Abstract: We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded. This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i. We then present some applications of this result.

• March 12

  Ivan Corwin (Microsoft Research New England and MIT)

  Macdonald processes and solvability in the KPZ universality class

  Abstract: The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.

• March 19

  Samuel Watson (MIT)

  Extreme nesting in the conformal loop ensemble

  Abstract: We compute the almost-sure Hausdorff dimension of the set of points surrounded by more or fewer than the typical number of loops in a realization of the conformal loop ensemble. More precisely, this is the set of points z for which the number of loops surrounding the disk of radius r centered at z is (theta + o(1)) log(1/r), where theta is a nonnegative parameter. We will also discuss a generalization of this result in which the loops are assigned iid weights. Using a coupling of CLE with the Gaussian free field, this leads to a new way of understanding GFF extremes.

  Joint work with Jason Miller and David Wilson.

• March 26

  No Seminar (Spring Break)

• April 2

  --SPECIAL DOUBLE SEMINAR--

  First speaker:

  Sourav Chatterjee (New York University)

  Invariant measures and the soliton resolution conjecture

  Abstract: The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. I will present a theorem that proves a "statistical version" of this conjecture at mass-subcritical nonlinearity. The proof involves a combination of techniques from large deviations, PDE, harmonic analysis and bare hands probability theory.

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  Second Speaker:

  Igor Kortchemski (Universite Paris-Sud)

  The Brownian triangulation: a universal limit for random plane non-crossing configurations

  Abstract: We are interested in various models of random plane non-crossingconfigurations consisting of diagonals of convex polygons, such as uniformtriangulations, dissections, non-crossing partitions or non-crossingtrees. For all these models, we prove convergence in distribution towardsAldous' Brownian triangulation of the disk. We will discuss interestingcombinatorial applications concerning the length of the longest diagonalor the maximal vertex degree. This is joint work with Nicolas Curien.

• April 9

  Julien Dubedat (Columbia University)

  Dimers observables and Ising correlations.

  Abstract: We discuss the asymptotic analysis of vertex observables of dimer height functions and their relations to correlators in the Ising model.

• April 17

  -- TUESDAY! (because Monday is Holiday)-- (usual room 2-135 at the usual time 4:15-5:15)

  Senya Shlosman (CPT)

  Rotating states

  Abstract: I will explain the construction of an interacting particle system which has a unique stationary state but which is not ergodic. In other words, the dynamics started from different initial states never converge to each other, while the stationary state of the dynamics is unique. The result can be interpreted as the case of the non-uniqueness of the steady state. Joint work with Christian Maes.

• April 23

  --SPECIAL DOUBLE SEMINAR--

  First speaker:

  Michael Aizenman (Princeton University)

  Does the 2D Random Field Ising Model exhibit a phase transition in the disorder parameter?

  Abstract: It is a know theorem that in low dimensional models of statistical mechanics, such as the Random Field Ising Model (RFIM) first order phase transitions are unstable with respect to the introduction of arbitrarily weak disorder in the field conjugate to the order parameter. The result (derived jointly with J. Wehr) will be explained in the lecture, along with its more recent generalization to quantum systems (derived jointly with R. Greenblatt and J. Lebowitz). The talk will however focus on the question whether there is, nevertheless, a phase transition in the the behavior of RFIM as function of the disorder strength. Its manifestation could be power law decay of correlations at weak disorder, which at high disorder changes to exponential decay. It is proposed that a relevant point of reference for the scaling limit of the ground state's ``sensitivity percolation'' could be Mandelbrot's percolation model. (Work in progress with Jack Hanson)

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  Second speaker:

  Remi Rhodes (Universite Paris-Dauphine)

  Generalized Gaussian multiplicative chaos

  Abstract: In a pioneering work, J.P. Kahane introduced the theory of Gaussian Multiplicative Chaos. Such an object is roughly speaking a random measure with density w.r.t. the Lebesgue measure given by the exponential of a Gaussian log-correlated field up to a multiplicative factor, called the intermittency parameter. When the intermittency parameter is small enough, these measures are non trivial and are characterized by a functional equation, which we call star-equation. Beyond some critical value, these measures vanish. However the star equation admits non trivial solutions, which we call generalized Gaussian multiplicative chaos. It turns out that Gaussian multiplicative chaos and generalized Gaussian multiplicative chaos satisfy a duality relation together with the KPZ formula, giving precise mathematical meaning to the KPZ duality introduced by theoretical physicists.

• April 30

  Chuck Newman (Courant Institute, NYU)

  Critical Scaling Limits and Measure Ensembles

  Abstract: In statistical physics, systemslike percolation and Ising models are of particular interestat their critical points. Critical systems have long-rangecorrelations that typically decay like inverse powers. Theircontinuum scaling limits are expected to have universal dimension-dependent properties.Critical two-dimensional scaling limitshave been studied by Schramm, Lawler, Werner, Smirnov, Sheffieldand others with a focus on the boundaries of large clusters. In thescaling limit these can be described by Schramm-Loewner Evolution(SLE) curves.

  In this talk, I'll discuss a different but related approach,which focuses on cluster area measures. In the case of thetwo-dimensional Ising model, this leads to a representation ofthe continuum Ising magnetization field in terms of sums ofcertain measure ensembles with random signs. This is basedon joint work with F. Camia and on work in progresswith F. Camia and C. Garban.

• May 7

  No Seminar because of the Simons Lectures

• May 14

  Omer Angel (University of British Columbia)

  Linearly reinforced random walks.

  Abstract: We prove that the linearly reinforced random walk on a bounded degree graph with sufficiently small initial weights is recurrent. On non-amenable graphs we also establish transience for sufficiently large initial weights. The vertex reinforced jump process is also treated. Join with Nicholas Crawford and Gady Kozma.

Links to Probability related conferences and programs:

• September 2011 - September 2012

  Symposium on Probability, Warwick UK

• January - May, 2012

  Random Spatial Processes, MSRI Berkeley California

• January 9-13, 2012

  France Winter School, Marne-la-Vallee

• March 12-16, 2012

  Arizona School of Analysis and Mathematical Physics, Tucson Arizona

• March 26-30, 2012

  XII Latin American Congress of Probability and Mathematical Statistics, Vina del Mar, Chile

• April 27-29, 2012

  Graduate Student Probability Conferene, Atlanta Georgia

• June 18-29, 2012

  St. Petersburg School in Probability and Statistical Physics, St. Petersburg Russia

• June 18-29, 2012

  Advances in Random Matrix Theory, IMA (Minnesota)

• June 18 - August 15, 2012

  Random Matrix Theory and its Application II, Singapore

• July 9-13, 2012

  8th World Congress in Probability and Statistics, Istanbul Turkey

• July 23-27, 2012

  Probability at Warwick Young Researchers Workshop

• August 6-11, 2012

  International Congress on Mathematical Physics, Aalborg Denmark