MIT Probability Seminar - Spring 2013

Schedule:

• February 4: David Wilson (Microsoft Research)

  Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs

  We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the ``intensity'' of the loop-erased random walk in Z^2, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a 15-year old conjecture about the stationary sandpile density on Z^2. We do the analogous computation for the triangular lattice, honeycomb lattice and Z * R, for which the probabilities are 5/18, 13/36, and 1/4-1/pi^2 respectively. Joint work with Richard Kenyon.
  
  
  

• February 11: Vadim Gorin (MIT)

  Finite-dimensional universality of GUE

  GUE is the distribution of eigenvalues of the random N*N Hermitian matrix with Gaussian entries. Asymptotic properties of this distribution as N tends to infinity attracted lots of attention and many of those properties turn out to be universal, in the sense that they are also present in the wide class of models of statistical mechanics and random matrix theory. In the talk we will discuss another kind of universality; I will try to explain that the GUE distribution for a finite N is a universal object itself. For N=1 this is well-known, since in this case GUE is nothing else, but Gaussian distribution. I will show that for general N the GUE distribution appears as a scaling limit in a large variety of probabilistic models on interlacing particle configurations.
  
  
  

• February 13: Harvard-MIT Random Matrix Theory Afternoon

  Speakers:

  Charles Bordenave (Toulouse)
  Patrick Ferrari (Bonn)
  Benjamin Schlein (Bonn)

• February 18: President's day - No seminar!

• February 25: No seminar!
  
  
  

• March 4: Nicolas Curien (ENS Paris)

  Random stable looptrees

  In this talk, we introduce a new class of random compact metric spaces which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be viewed, in a certain sense, as dual graphs of stable Lévy trees. We conjecture that they naturally arise as universal scaling limits of cluster boundaries in random planar maps decorated with an $O(n)$ model. We prove this conjecture for site-percolation on random triangulations. Our method also enables us to derive new properties (critical exponents...) of site percolation on the UIPT by giving a precise description of the external boundary of percolation clusters. Based on joint work in progress with Igor Kortchemski.
  
  
  

• March 11: Reda Chhaibi (Zurich)

  The geometric Robinson-Schensted correspondence and the Whittaker process

  The Whittaker process can be seen as a positive temperature analogue of Brownian motion in the Weyl chamber - Dyson’s Brownian motion for type A. In this talk, I will try to explain how it can be obtained through a geometric Robinson-Schensted correspondence. Also, I want to give formulas for Whittaker functions that use a canonical measure given by Brownian motion on the group.
  
  
  

• March 11-13: Norbert Wiener Lectures at Tufts, delivered by Peter Winkler (Dartmouth)
  
  
  

• March 18: -DOUBLE SEMINAR-
  
  First speaker 4:15-5:05: Dana Randall (Georgia Tech)

  Slow Mixing for the Hard-­Core Model on Z^2

  The hard-core model has attracted much attention across several disciplines, representing lattice gases in statistical physics and independent sets in discrete mathematics and computer science. On finite graphs, we are given a parameter $\lambda$, and an independent set $I$ arises with probability proportional to $\lambda^{|I|}$. We are interested in determining the mixing time of local Markov chains that add or remove a small number of vertices in each step. On finite regions of $Z^2$ it is conjectured that there is a phase transition at some critical point $\lambda_c$ that is approximately $3.79$. It is known that local chains are rapidly mixing when $\lambda < 2.3882$. We give complementary results showing that local chains will mix slowly when $\lambda > 5.3646$ on regions with periodic (toroidal) boundary conditions and when $\lambda > 7.1031$ with non-periodic (free) boundary conditions. The proofs use a combinatorial characterization of configurations based on the presence or absence of fault lines and an enumeration of a new class of self-­avoiding walks called taxi walks.

  Second speaker 5:10-6:00: Mykhaylo Shkolnikov (Berkeley)

  Large deviations for diffusions interacting through their rank

  Systems of diffusion processes (particles) with rank-based interactions have been studied heavily due to their importance in stochastic portfolio theory and the intriguing relations with particle systems appearing in statistical physics. We will study the behavior of this particle system as the number of particles gets large. By obtaining a large deviations principle, we will show that the limiting dynamics can be described by a porous medium equation with convection, whereas paths of finite rate are given by solutions of appropriately tilted versions of this equation. This is the first instance of a large deviations principle for diffusions interacting both through the drift and the diffusion coefficients with the diffusion coefficients not being globally Lipschitz (and not even continuous). Based on joint work with A. Dembo, S.R.S. Varadhan and O. Zeitouni.
  
  
  

• March 25: No seminar!
  
  
  

• April 1: Paul Fendley (Virginia)

  Free fermions and parafermions

  Some important models of statistical mechanics, such as dimers and the Ising model in two dimensions, can be "solved" by rewriting the degrees of freedom in terms of free fermions, i.e. expressing the transfer matrix in terms of the generators of a Clifford algebra. I will review these results, and describe how they can be generalized to models with Z_n symmetry. In particular, I show how to express the degrees of freedom in terms of certain combinations of parafermions, Z_n generalizations of fermions.
  
  
  

• April 8: -DOUBLE SEMINAR-
  
  First speaker 4:15-5:05: Jonathan Mattingly (Duke)

  Some modest examples of Stochastic stabilization

  Second speaker 5:10-6:00: Nike Sun (Stanford)

  The replica symmetry breaking solution to the maximum independent set problem

  Satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. These problems also arise naturally in combinatorics, in the study of sparse random graphs. While the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d exceeding a constant d(0), there exist explicit constants A, C depending on d such that the maximum size has constant fluctuations around A*n-C*(log n) establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. As an application of our method we also prove an explicit satisfiability threshold in random regular k-NAE-SAT. This is joint work with Jian Ding and Allan Sly.
  
  
  

• April 17: Jeremie Bouttier (CEA and ENS) Wednesday. Room 4-153

  Distances in random planar maps and discrete integrability

  Metric properties of random maps (graphs embedded in surfaces) have been subject to a lot of recent interest. In this talk, I will review a combinatorial approach to these questions, which exploits bijections between maps and some labeled trees. Thanks to an unexpected phenomenon of "discrete integrability", it is possible to enumerate exactly maps with two or three points at prescribed distances, and more. I will then discuss probabilistic applications to the study of the Brownian map (obtained as the scaling limit of random planar maps) and of uniform infinite planar maps (obtained as local limits). If time allows, I will also explain the combinatorial origin of discrete integrability, related to the continued fraction expansion of the so-called resolvent of the one-matrix model. Based on joint works with E. Guitter and P. Di Francesco.
  
  
  

• April 22: Hugo Duminil-Copin (Geneva)

  Parafermionic observables in planar Potts models and self-avoiding walks

  In this talk, we will discuss the role of parafermionic observables in the study of several planar statistical physics models. These objects have been introduced recently in order to prove conformal invariance of the Ising model. We will explain how they can be combined with combinatorial and probabilistic arguments to compute the connective constant for self-avoiding walks (the n=0 loop O(n)-model) on the hexagonal lattice, and to provide information on the critical phase of the Fortuin-Kasteleyn percolation (the graphical representation of Potts models). As an application of their use for FK percolation, we will show the absence of spontaneous magnetization for the critical planar Potts models with 2, 3 and 4 colors, thus proving part of the conjecture asserting that the planar Potts models undergo a discontinuous phase transition if and only if the number of colors is greater than 4.
  
  
  

• April 26: Harvard-MIT Random Matrices and Random Geometry Afternoon

  Speakers:

  Mark Rudelson (Michigan)
  Nikolai Makarov (Cal-Tech)
  Jean-Francois Le Gall (Paris-Sud Orsay)

• April 29: Jean-Francois Le Gall (Paris-Sud Orsay)

  The Brownian map: A universal limit for large random planar maps

  
  
  
  

• May 6: Greg Lawler (Chicago) 3:15-4:15. Room 2-136

  Minkowski content and the Schramm-Loewner evolution (SLE)

  The Schramm-Loewner evolution (SLE_kappa) is a measure on two-dimensional curves that arises as scaling limits in statistical physics. The Hausdorff dimension of the curves is d = 1 + min(1,kappa/8). The curve is usually parametrized by capacity but it is more natural to use a fractal parametrization that should correspond to the length of the curve. We prove that for d < 2, the d-dimensional Minkowski content of the curve exists and gives the natural length as defined previously by Scott Sheffield and myself. In particular, the natural length is reversible and independent of domain. This is joint work with Mohammad Rezaei.
  
  
  

• May 13: Raj Rao Nadakuditi (Michigan)

  Random matrix theory and perfect transmission in random media

  We consider the scientific problem of when and whether light can be perfectly transmitted through an "opaque" random medium such as white paper or eggshells. We develop a numerically rigorous, high-accuracy solver to investigate the statistics of the associated random scattering matrices and observe that perfect transmission is (almost) always possible. Experiments with the numerical solver allow us to develop a basic yet general random matrix model for this problem. Using free probability theory to flesh out this connection brings into focus a closed-form expression for the transmission coefficient distribution that predicts when perfect transmission is possible. The predicted distribution matches the experimental statistics unreasonably well suggesting new avenues for extending free probability theory. This is joint work with Curtis Jin and Eric Michielssen.
  
  
  

Semester/Year programs:

September - December 2012: Institute for Computational and Experimental Research in Mathematics Semester Program on Computational Challenges in Probability, Providence RI

April - September 2013: Lebesgue Center Semester Program on Perspectives in Analysis and Probability, Rennes France

September 2013 - June 2014: Institute for Advanced Studied Year Program on Non-equilibrium Dynamics and Random Matrices

May 27-31, 2013: IHP Spring School on Threshold phenomena and random graphs, Paris, France.

June 3-7, 2013: Lebesgue Center Summer School on KPZ Equation and Rough Paths, Rennes France

June 3-15, 2013: The Beg Rohu Summer School on Disordered Systems, Saint Pierre Quiberon, France

July 14-26, 2013: 9th Cornell Probability Summer School , Cornell

July, 2013: Fields Institute focus program on Noncommutative Distributions in Free Probability Theory , Toronto, Canada

August 4-10, 2013: 17th Brazilian School of Probability

August 5-19, 2013: Bielefeld University Summer School on Randomness in Physics and Mathematics

March 14-16, 2013: Seminar on Stochastic Processes, Duke

July 29, 2013 - August 2, 2013: Stochastic Processes and Applications, Boulder, CO

July 22-26, 2013: StatPhys 25, Seoul, Korea

August 19-23, 2013: Analysis of Stochastich Partial Differential Equations, Michigan State University

Probability seminars in past semesters: