Schedule

• September 19

  Jian Ding (Chicago)

  Random planar metrics of Gaussian free fields

  Abstract: I will present a few recent results on random planar metrics of two-dimensional discrete Gaussian free fields, including Liouville first passage percolation, the chemical distance for level-set percolation and the electric effective resistance on an associated random network. Besides depicting a fascinating picture for 2D GFF, these metric aspects are closely related to various models of planar random walks.Based on joint works with Marek Biskup, Alexander Dunlap, Subhajit Goswami, Li Li and Fuxi Zhang.

• September 26

  Stéphane Benoist (MIT)

  Conformally invariant loop measures

  Abstract: We will discuss several aspects of a conjecture by Kontsevich and Suhov regarding existence and uniqueness of a one parameter family of conformally invariant measures on simple loops, conjecturally related to the SLE curve family. The most natural case (zero central charge i.e. SLE parameter kappa=8/3) was understood in a paper of Werner predating the conjecture. In a work in progress, Dubédat and myself construct loop measures in the whole conjectural range of existence (i.e. parameters kappa for which SLE is a simple curve) by using imaginary geometry techniques.

• September 30

  Friday

  Charles River Lectures

• October 3

  Michał Kotowski (Toronto)

  Limits of random permuton processes and large deviations for the interchange process

  Abstract: We will present results on large deviations for the interchange process on a line. The study is done in the framework of so-called permuton processes, which provide a notion of a limit for permutation-valued stochastic processes. Our motivation comes from the analysis of random sorting networks and I will explain how the random sine curve process and methods from interacting particle systems enter the picture. Joint work with Balint Virag.

• October 10

  Columbus Day

• October 17

  Benoît Laslier (Cambridge)

  Universality of height fluctuations in the dimer model

  Abstract: I will present a new approach to the study of height fluctuations in the dimer model, not writing any determinantal formula. The key idea will be to make precise the fact that the imaginary geometry coupling between SLE and GFF is the continuous limit of the bijections between dimers and spanning, and to use the convergence of loop-erased random walk to SLE. We will obtain convergence of the height in many settings, including lozenge tilings of "Temperleyan like" planar domains of arbitrary slope and global shape.

• October 24

  Fabio Toninelli (Lyon)

  A (2+1)-dimensional growth process for discrete interfaces: stationary states, fluctuations and hydrodynamic limit

  Abstract: I will talk about a Markov chain on lozenge tilings of the plane, introduced by A. Borodin and P. L. Ferrari [CMP 2014]. This can be viewed as a 2+1-dimensional stochastic growth process (the growing discrete interface being the height function associated to the tiling) or as a totally asymmetric interacting 2-d particle system. I will briefly recall some results from Borodin and Ferrari, and then present new results on stationary states, growth of fluctuations and hydrodynamic limit. This is based on arxiv:1503.05339 and on work in progress with M. Legras.

  If time allows, I will present related results obtained with A. Borodin and I. Corwin

• October 31

  Horng-Tzer Yau (Harvard)

  The two-dimensional Coulomb plasma

  Abstract: For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order N, the number of particles of the gas, with an effective error bound N^{1-\kappa} for some constant \kappa > 0. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature.

• November 7

  Vadim Gorin (MIT)

  Universal asymptotic behavior of discrete particle systems beyond integrable cases

  Abstract: The talk is about a class of systems of 2d statistical mechanics, such as random tilings, noncolliding walks, log-gases and random matrix-type distributions. Specific members in this class are integrable, which means that available exact formulas allow delicate asymptotic analysis leading to the Gaussian Free Field, sine-process, Tracy-Widom distributions. Extending the results beyond the integrable cases is challenging. I will speak about a recent progress in this direction: about universal local limit theorems for a class of lozenge and domino tilings, noncolliding random walks; and about GFF-type asymptotic theorems for global fluctuations in these systems and in discrete beta log-gases.

• November 14

  Van Vu (Yale)

  How many real roots does a random polynomial have?

  Abstract: We will first survey several famous works concerning this fascinating question, following the footsteps of Waring, Silvester, Littlewood-Offord, Kac, Erdos, Ibragimov and many others. The rest of the talk is devoted to a new method that we have been developing with T. Tao in the last few years, which leads to improvements upon many classical results and solutions to a variety of open questions.

• November 21

  Jonathan Novak (UC San Diego)

  Lozenge tilings, orbital integrals, Hurwitz numbers

  Abstract: The connection between random lozenge tilings of planar domains and characters of general linear groups is well understood. Less appreciated is that there is a choice to be made in how this connection is applied to obtain limit theorems. Practitioners of integrable probability have always selected the Weyl character formula as their preferred tool. I will explain what happens if one chooses the Kirillov character formula instead.

• November 28

  Christopher Hoffman (University of Washington)

  Geodesics in First Passage Percolation

  Abstract: We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. We will present a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Gerandy Brito and Daniel Ahlberg.

• December 5

  Evgeni Dimitrov (MIT)

  GUE corners process in the six-vertex model

  Abstract: The talk is about a class of probability distributions on the six-vertex model, which originate from the higher spin vertex models of Borodin and Petrov . For this class of distributions there are operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. For certain boundary conditions, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a fixed choice of parameters this method can be used to prove that the asymptotic behavior of a six-vertex model near the boundary is described by the GUE-corners process.

• December 12

  Thomas Leblé (NYU)

  Microscopic behaviour of one- and two-dimensional log-gases

  Abstract: I will present results obtained with S. Serfaty concerning the behaviour of log-gases (or beta-ensembles) at arbitrary temperature in one and two dimensions. d=1 corresponds to eigenvalues of random Hermitian matrices and d=2 is known as the ''one-component plasma''in the physics literature.

  Using the 'next-order energy' introduced by Sandier-Serfaty, we characterise the microscopic arrangement of the particles through a large deviation principle whose rate function is weighted by the temperature - minimisers include the Sine-beta processes and the Ginibre point process.

  For d=1 we study the limits of high and low temperature and recover the fact that Sine-beta interpolates between a Poisson process and a perfect lattice.

  For d=2 we prove local laws and a central limit theorem for the fluctuations of linear statistics.