Schedule

• September 10

  -DOUBLE SEMINAR-

  First speaker 4:15-5:15

  Alan Hammond (Oxford)

  Self avoiding walk is sub-ballistic

  Abstract: In a joint work with Hugo Duminil-Copin, we prove that self-avoiding walk on Z^d is sub-ballistic in any dimension d at least two. That is, writing ||u|| for the Euclidean norm of u \in Z^d, and SAW_n for the uniform measure on $n$-step nearest-neighbour self-avoiding walks gamma that start at the origin of Z^d, we show that, for each v > 0, there exists c > 0 such that, for each positive integer n, SAW_n ( || gamma_n || > v n) < e^{- c n}.

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  Second speaker 5:30-6:30

  Leonid Petrov (Northeastern)

  Random lozenge tilings of polygons and their asymptotic behavior

  Abstract: I will discuss the model of uniformly random tilings of polygons drawn on the triangular lattice by lozenges of three types (equivalent formulations: dimer models on the honeycomb lattice, or random 3-dimensional stepped surfaces). Asymptotic questions about these tilings (when the polygon is fixed and the mesh of the lattice goes to zero) have received a significant attention over the past years.

  Using new formula for the determinantal correlation kernel of the model, I manage to establish the conjectural local asymptotics of random tilings in the bulk (leading to ergodic translation invariant Gibbs measures on tilings of the whole plane), and the predicted behavior of interfaces between so-called liquid and frozen phases (leading to the Airy process). Bulk asymptotic behavior allows to reconstruct the limit shapes of random stepped surfaces obtained by Cohn, Propp, Kenyon, and Okounkov. We also prove a conjecture of Kenyon (2004) that the large-scale asymptotics of random tilings are asymptotically governed by the Gaussian Free Field.

  As a particular case, all my results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

• September 17

  -DOUBLE SEMINAR-

  First speaker 4:15-5:05

  Charles Smart (MIT)

  Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity.

  Abstract: (Joint work with Scott N. Armstrong.) We prove regularity and stochastic homogenization of certain nonlinear degenerate ellipticequations in nondivergence form. In probabilistic terms, our resultscorrespond to quenched invariance principles for controlled diffusionprocesses in random environments. Treating such nonlinear equationsis difficult due to the absence of invariant measures.

  We allow the ellipticity to oscillate on the microscopic scale andassume only that its $d$th moment, where $d$ is the dimension, isbounded. In the general stationary-ergodic framework, we show thatthe equation homogenizes to a deterministic, uniformly ellipticequation. Showing that such an equation behaves like a uniformlyelliptic equation requires a novel reworking of the regularity theory. We also show that the moment condition is sharp by giving an explicitexample of an equation whose ellipticity has a finite $p$th moment,for every $p < d$, but for which regularity and homogenization breakdown. Some of our results are new even for linear equations.

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  Second speaker 5:10-6:00

  Anirban Basak (Stanford)

  Ferromagnetic Ising measures on large locally tree-like graphs

  Abstract: Consider the ferromagnetic Ising measure on sparse finite graphs converging locally to limiting tree ${\sf T}$. In case ${\sf T}$ is $d$-regular, it was recently shown by Montanari, Mossel, and Sly that these Ising measures converge locally to symmetric mixture of plus and minus (boundary conditions) Ising measures on ${\sf T}$, and for expander graphs, conditioned on positive magnetization these measures converge to plus-boundary condition Ising measure on ${\sf T}$. With Amir Dembo, we extend these results, and show universality for a more general, random limiting tree ${\sf T}$. In this talk I will review the results of Montanari et al. and discuss the results we obtain.

• September 24

  Yashodhan Kanoria (MSR New England)

  The Set of Solutions of Random XORSAT Formulae

  Abstract: The XOR-satisfiability (XORSAT) problem requires finding an assignment of n Boolean variables that satisfies m exclusive OR (XOR) clauses, where each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing n variables and m clauses of size k each. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as k-satisfiability (k-SAT). For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random k-XORSAT. In particular we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is achieved through a low complexity iterative algorithm.

  Based on joint work with Morteza Ibrahimi, Matt Kraning and Andrea Montanari.

• October 1

  Rick Kenyon (Brown)

  The hexahedron recurrence and the Ising model

  Abstract: This is joint work with Robin Pemantle.Two recurrence relations which are fundamental in combinatorics,integrable systems, and statistical mechanics are the "octahedron recurrence" also known as Hirota's bilinear difference equation, and "cube recurrence" or Miwa equation. We introduce a cousin of these, dubbed the "hexahedronrecurrence", show how it is related to the Ising model,and in particular show how this uncovers a cluster algebra structure in the Ising model.

• October 5

  Special full day event!

  Charles River Lectures on Probability and Related Topics

  Detail

  The Charles River Lectures on Probability and Related Topics will be hosted by Microsoft Research New England. The lectures are jointly organized by Harvard University, Massachusetts Institute of Technology 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:

  • Alice Guionnet (Massachusetts Institute of Technology)
  • Gerard Ben Arous (New York University)
  • Irit Dinur (Weizmann Institute of Science)
  • Persi Diaconis (Stanford University)
  • Yuval Peres (Microsoft Research)

  Details on how to register as well as directions to Microsoft Research are on the conference website

  Event organizers:

  • Alexei Borodin (Massachusetts Institute of Technology)
  • Ivan Corwin (Clay Mathematics Institute and Massachusetts Institute of Technology)
  • Yashodhan Kanoria (Microsoft Research New England)
  • Jason Miller (Massachusetts Institute of Technology)
  • Scott Sheffield (Massachusetts Institute of Technology)
  • H.T. Yau (Harvard University)

  For questions regarding the event, please contact Ivan Corwin.

• October 9

  Tuesday. Room 2-139

  Konstantinos Spiliopoulos (Boston University)

  Large Deviations, Metastability and Monte Carlo Methods for Multiscale Problems

  Abstract: We discuss large deviations, metastability and Monte Carlo methods for multiscale dynamical systems that are stochastically perturbed by small noise. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Using stochastic control arguments we identify the large deviations principle for each regime of interaction. The large deviations principle can then be used to study metastability for such problems, as well as asymptotic problems for related PDE's. Furthermore, we derive a control (equivalently a change of measure) that allows to design asymptotically efficient importance sampling schemes for the estimation of associated rare event probabilities and expectations of functionals of interest. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies. Time permitting we will discuss construction of efficient Monte Carlo methods for the related problem of escape from an attractor.

• October 16

  Tuesday. Room 2-139

  Amir Dembo (Stanford University)

  Persistence probabilities

  Abstract: Persistence probabilities concern how likely it is that a stochasticprocess has a long excursion above fixed level and of what are the relevant scenarios for this behavior. Power law decay is expectedin many cases of physical significance and the issue is to determineits power exponent parameter. I will survey recent progress in thisdirection (jointly with Jian Ding, Fuchang Gao, and Sumit Mukherjee),dealing with random algebraic polynomials of independent coefficients, iterated partial sums and other auto-regressivesequences, and with the solution to heat equation initiated bywhite noise.

• October 16-18

  Eisenbud lecture series in Mathematics and Physics delivered by Craig Tracy at Brandeis University

• October 22

  Ofer Zeitouni (University of Minnesota and Weizmann Institute)

  From KPP equations to the maxima of Gaussian free fields

  Abstract: The Fisher-Kolmogorov-Petrovsky-Piscounov equation describes the propagation of a self interacting wave such as in flame propagation. Probabilistically,it describes the evolution of the location of the maximal particle in a system of branching random walks (where the total population sizeincreases exponentially). That particle system was analyzed by Bramson using a mixture of probabilistic and analytic methods in the early 80's.

  The Gaussian free field is a random Gaussian field that is indexed by points in $R^d$; in the critical dimension d=2, it represents a random distribution.A discrete analogue of the GFF can be defined on any finite graph. Of special interest are properties of the maxima of the field, and in particular thefluctuations of the maximum.

  I will describe recent work that links the two objects, branching random walks and Gaussian free fields, in the critical dimension $d=2$. In particular, I willexplain how Bramson's work can be adapted in order to show that the maximum of the two dimensional (discrete) GFF has fluctuations which are of order $1$.

  The talk is based on joint works with Maury Bramson and with Jian Ding.

• October 29

  Seminar canceled! (Greg Lawler's talk has been canceled due to inclement weather and Soumik Pal's has been rescheduled for October 31.)

• October 31

  *RESCHEDULED* Wednesday. Room 2-142 at 3:00pm

  Soumik Pal (University of Washington)

  Eigenvalues of sparse random regular graphs

  Abstract: Adjacency matrices of sparse random regular graphs are longconjectured to lie within the universality class of random matrices.However, there are few rigorously known results. We focus onfluctuations of linear eigenvalue statistics of a stochastic processof such adjacency matrices growing in dimension. The idea is tocompare with eigenvalues of minors of Wigner matrices whosefluctuation converges to the Gaussian Free Field. We show that lineareigenvalue statistics can be described by a family of Yule processeswith immigration. Certain key features of the Free Field emerge as thedegree tends to infinity. Based on joint work with Tobias Johnson.

• November 5

  David Gamarnik (MIT)

  Graph limits, large deviations and algorithms for sparse graphs.

  Abstract: The theory of converging graph sequences is well developed for the class of dense graphs. The theory of converging sparse graph sequences, however, is far less understood. We show that prior definitions of converging sparse graph sequences are inadequate to captureimportant graph theoretic properties, and introduce a new definition based on thelarge deviations theory. We show that the new definition implies most the known types of convergences and conjecturethat sparse random graphs are converging in the sense of the new definition. Establishing this conjecture will havean important implications for the theory of spin glasses. In the second part of the talk we will discuss algorithmic hardness of solving combinatorial optimization problems on sparse random graphsusing the so-called local algorithms. Recently Hatami, Lovasz and Szegedi proposed a particular framework for local algorithmsand conjectured that such algorithms exist for the problem of finding a largest independent set in a random regular graph.We disprove this conjecture by showing that no local algorithm is capable of producing an independent set larger thatfactor 1/2 of the optimal. Our result is based on a powerful clustering phenomena discovered by statistical physicistsin the context of spin glass theory, and recently confirmed by rigorous methods. To the best of our knowledge,our result is the first direct application of the statistical physics methods to the area of algorithmic hardness. Joint work with Jennifer Chayes, Christian Borgs and Madhu Sudan.

• November 13

  Tuesday. Room 2-139 at 3:00pm *SPECIAL TIME*

  Pierre van Moerbeke (Universite Catholique de Louvain and Brandeis University)

  Tacnode processes in Aztec diamonds

  Abstract: Two different determinantal point process and their kernels can be associated in a natural way to two overlapping Aztec diamonds (double Aztec diamonds). When the size of diamond gets very large, keeping the overlap finite and fixed, the limit of these kernels, for an appropriate scaling, leads to a new kernel, which can be viewed as a coupling two GUE-minor kernels; this happens when the tacnode for the boundary of the frozen region (for the double Aztec diamonds) is macroscopically near the edge of the diamonds. This leads in particular to a finite rank deformation of GUE. This talk is joint work with Mark Adler, Sunil Chhita and Kurt Johansson.

• November 19

  -DOUBLE SEMINAR-

  First speaker 4:15-5:05:

  Jian Ding (University of Chicago)

  Markov type and the multi-scale geometry of metric spaces

  Abstract: It is now well-understood that the behavior of Markov chains in metricspaces can be used to capture many facets of their geometric structure. In1992, K. Ball introduced the Markov type of a metric space which measuresthe rate of drift of reversible chains in the space. This notion has sinceseen many geometric applications.While Markov type is a bi-Lipschitz invariant, there were a number ofspaces conjectured to have Markov type 2 that do not bi-Lipschitz embedinto any known space with Markov type 2. We show that, at least forembeddings into L_p spaces, a much weaker sort of embedding suffices. Thisallows us to exhibit Markov type 2 for many spaces, including planar graphmetrics and doubling metrics, answering questions of Naor, Peres, Schramm,and Sheffield (2004). The main technical obstacle involves jointlycontrolling a family of martingales whose difference sequences are alldominated by a single random variable.This is joint work with James Lee and Yuval Peres.

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  Second speaker 5:10-6:00:

  Subhroshekhar Ghosh (University of California, Berkeley)

  What does a Point Process Outside a Domain tell us about What's Inside?

  Abstract: In a Poisson point process we have independence between disjoint spatialdomains, so the points outside a disk give us no information on the pointsinside. The story gets a lot more interesting for spatially correlatedprocesses. We focus on the two main natural examples ofrepulsive point processes on the plane - the Ginibre ensemble (arisingfrom eigenvalues of random matrices) and zero ensembles of certainGaussian power series. We show that here the outside points actually tellus a lot - they determine almost surely the "mass" or the "centre of mass"of the inside points (as the case may be), and that they determine"nothing more".This gives us a glimpse into a hierarchy of point processes based on theirrigidity, of which we know only the simplest examples.Time permitting, we will also look at several interesting consequences ofour results, with applications to continuum percolation, reconstruction ofGaussian entire functions, completeness of random exponentials, andothers.

• November 26

  -DOUBLE SEMINAR-

  First speaker 4:15-5:05:

  Paul Bourgade (Harvard University)

  Mesoscopic analogies between L-functions and random matrices.

  Abstract: Fluctuations of random matrix theory type have been known to occur in analytic number theory since Montgomery's calculation of the pair correlation of the zeta zeros, in the microscopic regime. At the mesoscopic scale, the analogy holds, through a limiting Gaussian field, which present an ultrametric structure similar to log-gases. In particular we will consider an analogue of the strong Szeg{\H o} theorem for L-functions.

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  Second speaker 5:10-6:00:

  Clement Hongler (Columbia University)

  Conformal Invariance of Ising Model Correlations

  Abstract: We review recent results with Dmitry Chelkak and Konstantin Izyurov, where we rigorously prove existence and conformal invariance of scaling limits of magnetization and multi-point spin correlations in the critical Ising model on an arbitrary simply connected planar domain. This solves a number of conjectures coming from physical and mathematical literatures. The proof is based on convergence results for discrete holomorphic spinor observables and correlation inequalities.

• December 3

  Eyal Lubetzky (Microsoft Research)

  The shape of (2+1)-dimensional SOS

  Abstract: We present new results on the (2+1)-dimensional Solid-On-Solid model at low temperatures. Bricmont, El-Mellouki and Froelich (1986) showed that in the presence of a floor there is an entropic repulsion phenomenon, lifting the surface to a height which is logarithmic in the side of the box. We refine this and establish the typical height of the SOS surface is precisely the floor of [1/(4\beta)\log n], where n is the side-length of the box and \beta is the inverse-temperature. We determine the asymptotic shape of the top plateau and show that its boundary fluctuation are n^{1/3+o(1)}.

  Based on joint works with Pietro Caputo, Fabio Martinelli, Allan Sly and Fabio Toninelli.

• December 10

  (TBA)

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

• June 3-7, 2013

  Lebesgue Center Summer School on KPZ Equation and Rough Paths, Rennes 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

Conferences and Lecture Series

• October 5, 2012

  Charles River Lectures on Probability and Related Topics, Microsoft Research New England

• October 15-16, 2012

  Ahlfors Lecture Series, Elon Lindenstrauss, Havard

• October 16-18, 2012

  Eisenbud Lecture Series in Mathematics and Physics, Craig Tracy, Brandeis

• November 16-17, 2012

  Current Developments in Mathematics 2012, Harvard

• December 16-18, 2012

  108th Statistical Mechanics Conference, Rutgers

• 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