Schedule

• September 2

  Perla Sousi (Cambridge University)

  Mixing times are hitting times of large sets

  Abstract: We consider irreducible reversible discrete time Markov chains on a finite state space. Mixing times and hitting times are fundamental parameters of the chain. In this talk, we relate them by showing that the mixing time of the lazy chain is equivalent to the maximum over initial states x and large sets A of the hitting time of A starting from x. As an application, we show that the mixing time on a finite binary tree is robust to bounded change of edge conductances. (joint work with Yuval Peres)

• September 12

  Ivan Corwin (Microsoft Research and MIT)

  Brownian Gibbs line ensemble.

  Abstract: The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble.

  The KPZ line ensemble arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This talk will also introduce this line ensemble and demonstrate that it too has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost.

• September 19

  Charles Smart (MIT)

  The Fractal Nature of the Abelian Sandpile

  Abstract: I will discuss recent joint work with Lionel Levine (Cornell) and Wesley Pegden (NYU). The Abelian Sandpile is adiffusion process on configurations of chips on the integer lattice on \Z^d. The stabilized single-source sandpile has a distinctive imagewhich we now know has a continuum limit. Moreover, we can alsoexplain the fractal structure of the continuum limit.

• September 26

  Alex Bloemendal (Harvard)

  Finite rank perturbations of random matrices and Tracy-Widom(beta)

  Abstract: The top eigenvalues of finite rank perturbations of large Wigner andsample covariance matrices are known to exhibit a phase transition.In recent joint work with Balint Virag we show they have a limit nearthe transition, solving an outstanding problem in the real case.Bypassing joint densities, we identify a continuum operator limit.The resulting deformations of Tracy-Widom(beta) can be characterizedin terms of a hitting probability of a modified Dyson's Brownianmotion, and in terms of a related linear PDE; both feature beta as asimple parameter.

• October 3

  No Seminar in order not to overlap with Eliot Lieb's talk at Harvard (4.15-5:15 PM in SC Hall B).

• October 10

  No Seminar (Columbus Day)

• October 17

  --SPECIAL DOUBLE SEMINAR--

  First speaker 4:15-5:15:

  Van Vu (Yale)

  Random matrices: Getting inside the bulk !

  Abstract: Random matrix theory is a well developed area of mathematics with strong links to various other areas such as mathematical physics, probability, number theory, combinatorics, to mentioned a few. For a very long time, most results in the theory focus on the global distributions of the eigenvalues or the distribution of the eigenvalues at the edge of the spectrum (for example the Tracy-Widom law). Getting information about eigenvalues inside the bulk of the spectrum seemed out of reach and was done in very special cases, such as for matrices with gaussian entries.

  The situation changed substantially in the last few years, due to the works of Erdos et. al. and Tao and the speaker. In this talk, I am going to present a new method developed by Tao and myself. This method is motivated by the Lindenberg replacement method in probability theory and enabled us to get limiting distribution of every single eigenvalue in the spectrum. This gives a new way to attack many long standing problems. For instance, combining our method with recent results of Erdos et. al., we recently proved an old conjecture of Wigner-Dyson-Mehta on correlation functions in its full generality.

  ---

  Second speaker 5:30-6:30:

  Vladas Sidoravicius (IMPA)

  From random interlacements to coordinate and infinite cylinder percolation

  Abstract: During the talk I will focus on the connectivity properties of three models with long (infinite)range dependencies: Random Interlacements, percolation of the vacant setin infiniterod model and Coordinate percolation. The latter model have polynomialdecay in sub-critical and super-critical regime in dimension 3.I will explain the nature of this phenomenon and why it is difficult tohandle these models technically. In the second halfof the talk I will present key ideas of the multi-scale analysis whichallows to reach some conclusions. At the end I will discussapplications and several open problems.

• October 24

  Jim Propp (UMass Lowell)

  Rotor-routing, smoothing kernels, and reduction of variance: breaking the O(1/n) barrier

  Abstract: If a random variable X is easier to simulate than to analyze,one way to estimate its expected value E(X) is to generaten samples that are distributed according to the law of X and take their average. If the samples are independent, then (assuming X has finite variance) the estimate will have typical error O(1/sqrt(n)). But often one can do better by introducing appropriate forms of negative dependence. In the toy context of simulating Markov chains to estimate their absorption probabilities, I'll describe a scheme that uses maximally anticorrelatedidentically distributed Bernoulli random variables (aka rotorrouters) and has typical error O((log n)/n), and a related scheme with typical error O(1/n). This might seem to be optimal, and indeed one cannot expect the average (X_1+...+X_n)/n to differ from its expected value E(X) by less than O(1/n). However, by using weighted averages that assign X_i less weight when i is near 1 or n and greater weight when i is near n/2, one can get estimators for E(X) with typical error significantly smaller than O(1/n).

  The methods and ideas are mostly probabilistic and combinatorial. No prior knowledge of rotor-routing or smoothing kernels, and no familiarity with (or fondness for) statistics, will be assumed.

• October 31

  Mark Adler (Brandeis University)

  Random Matrix Theory Minors and Percolation Theory

  Abstract: Motivated by percolation theory, we give 3 new minor processes, and their correlation kernels. The first process has to do with GUE with an external source, while the second and third with Wishart and Jacobi with "gaps". These processes, are in distribution equivalent to certain percolation processes.The correlation kernel leads to showing that certain scaling limits of these processes lead to the universal Pearcey process of RMT. The original GUE minor process had to do with the joint distribution of the spectrum of the all the minors of an N by N GUE matrix and has now come up in many models, and so seems canonical in describing certain behavior and hopefully these processes will play a similar role.

• November 7

  Jonathan Novak (MIT)

  Monotone Hurwitz numbers and the HCIZ integral

  Abstract: It is well-known that the large N limit of the Hermitian one-matrix model is, for polynomial potentials, an analytic function in the coefficients of the potential whosepower series expansion is a generating function enumerating tessellations of a sphereby polygonal tiles of given shapes. Since the limiting free energy may be determined analytically bypushing forward onto the space of eigenvalues, one can use this connection to obtain exact counting formulas for planar maps.When one tries to repeat this process for the two-matrix model, the reduction to eigenvalues is impeded by a trickyintegral over the unitary group: the Harish-Chandra-Itzykson-Zuber integral. The HCIZ integral itself can be viewed asthe partition function of a Gibbs measure on the unitary group. I will discuss joint work with I. Goulden and M. Guay-Paquetin which we produce a combinatorial problem solved by the asymptotics of the HCIZ model. This problem is a variant ofthe classical Hurwitz problem, which asks for the number of branched covers of the sphere having given singular data. By solvingthe combinatorial problem directly, we are able to prove compact convergence of the free energy of the HCIZ model.

• November 14

  Amir Dembo (Stanford University)

  Potts and independent set models on d-regular graphs.

  Abstract: We consider the ferromagnetic Potts on typical d-regular graphs, and the independent set model on typical bipartite d-regular graphs,with graph size tending to infinity. We show that thereplica symmetric (Bethe) prediction applies forall parameter values in these two models.

  In this talk I will describe some of the proof techniques,which will give an indication of the contrast with anti-ferromagneticPotts model and the independent set model at high fugacity onnon-bipartite graphs, where the Bethe prediction is known to fail.

  This is joint work with Andrea Montanari, Allan Sly and Nike Sun.

• November 21

  No Seminar.

  Note that the conference Current Developments in Mathematics 2011 (Harvard, Nov. 18 - Nov. 19) will have wonderfull probability related talks by Richard Kenyon, Robin Pemantle, and Jeremy Quastel.

• November 28

  Jason Miller (Microsoft Research, Redmond)

  Imaginary Geometry and the Gaussian Free Field

  Abstract: The Schramm-Loewner evolution (SLE) is the canonical model of anon-crossing conformally invariant random curve, introduced by OdedSchramm in 1999 as a candidate for the scaling limit of loop erasedrandom walk and the interfaces in critical percolation. Thedevelopment of SLE has been one of the most exciting areas inprobability theory over the last decade because Schramm's curves havenow been shown to arise as the scaling limit of the interfaces of anumber of different discrete models from statistical physics. In thistalk, I will describe how SLE curves can be realized as the flow linesof a random vector field generated by the Gaussian free field, thetwo-time-dimensional analog of Brownian motion, and how thisperspective can be used to resolve a number of open conjecturesregarding the sample path behavior of SLE.

  Based on joint work with Scott Sheffield.

• December 5

  Antonio Auffinger (University of Chicago)

  A simplified proof of the relation between scaling exponents in first passage percolation

  Abstract: In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved a version of this conjecture using a strong definition of the exponents. I will discuss work I just completed with Michael Damron, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the scaling relation. One advantage of our argument is that it does not require a non-trivial technical assumption of Chatterjee on the weight distribution.

• December 12

  Gregory Miermont (Universite Paris-Sud)

  The scaling limit of random quadrangulations

  Abstract: I will present some recent progress on the convergence of rescaled large random quadrangulation -- i.e. a large uniform gluing of squares forming a topological sphere -- towards a continuum object called the Brownian map, which is a universal model for a continuum random surface. I will convey some of the main ideas of the proof, which requires a precise study of geodesics in large quadrangulations and in the limiting space, and in particular, of the locus where these geodesics tend to separate. If time allows I will also mention some questions concerning loop models on random quadrangulations.

• December 14

  Peter Sarnak (Princeton) is giving a joint combinatorics/probability talk at the 4.15-5.15 in room 2-135, entitled Randomness on the Mobius function.