Schedule

• February 13

  NO SEMINAR (snow)

• February 21

  TUESDAY, 4:15 -- 5:15, ROOM 4-257

  Martin Tassy (UCLA)

  Variational principles for discrete maps

  Abstract: Previous works have shown that arctic circle phenomenons and limiting behaviors of some integrable discrete systems can be explained by a variational principle. In this talk we will present the first results of the same type for a non-integrable discrete system: graph homomorphisms from Z^d to a regular tree. We will also explain how the technique used could be applied to other non-integrable models.

• February 27

  Shirshendu Ganguly (Berkeley)

  Diffusive estimates for random walk under annealed polynomial growth

  Abstract: We show that on a random infinite graph G of polynomial growth where simple random walk is stationary, it is diffusive along a subsequence, i.e., the second moment of the distance from the starting point grows at most linearly in time. This extends a result of Kesten that applied to the extrinsic metric on subgraphs of the lattice Z^d, and answers a question due to Benjamini, Duminil-Copin, Kozma and Yadin. We also show that, in general, passing to a subsequence is necessary. As a consequence, we deduce that harmonic functions of sublinear growth on such graphs G are constants. Our proof combines embeddings with the mass transport principle.

  Based on joint work with James Lee and Yuval Peres.

• March 6

  Curtis T McMullen (Harvard)

  Thermodynamics, Hausdorff dimension and the Weil-Petersson metric

• March 7

  TUESDAY, 4:15--5:15, ROOM 2-361

  Victor-Emmanuel Brunel (MIT)

  Likelihood geometry of determinantal point processes

  Abstract: Determinantal point processes (DPPs) have attracted a lot of attention in probability theory, because they arise naturally in many integrable systems. In statistical physics, machine learning, statistics and other fields, they have become increasingly popular as an elegant mathematical tool used to describe or to model repulsive interactions. In this talk, we study the geometry of the likelihood associated with such processes on finite spaces. Interestingly, the local behavior of the likelihood function can be very different according to the structure of a specific graph that we define for each DPP. Finally, we discuss some statistical consequences of this fact, namely, the asymptotic accuracy of a maximum likelihood estimator.

• March 13

  Nike Sun (Berkeley)

  Supercritical minimum mean-weight cycles

  Abstract: We consider the minimum mean-weight cycle (MMWC) in the stochastic mean-field distance model, that is, in the complete graph on n vertices with edges weighted by independent exponential random variables (of unit rate). Mathieu and Wilson [MW] (2012) showed that the MMWC exhibits very different characteristics according to whether its mean weight is smaller or larger than 1/(ne), where both cases occur with asymptotically positive probability. While the behavior in the subcritical (below 1/(ne)) regime is characterized in detail by [MW], much less was understood in the supercritical regime (above 1/(ne)). I will describe some of the obstacles, and present our results determining the length and weight asymptotics for the supercritical MMWC.

  Joint work with Jian Ding and David B. Wilson.

• March 20

  Fredrik Vikulnd (KTH)

  Convergence of loop-erased walk in the natural parametrization

  Abstract: Loop-erased random walk (LERW) is the random self-avoiding walk one gets after erasing the loops in the order they form from a simple random walk. Lawler, Schramm and Werner proved that LERW in 2D converges in the scaling limit to SLE(2) as curves viewed up to reparametrization. It is however more natural to view the discrete curve as parametrized by (renormalized) length and it has been believed for some time that one then has convergence to SLE(2) equipped with the so-called natural parametrization, which in this case is the same as 5/4-dimensional Minkowski content. I will discuss recent joint works with Greg Lawler (Chicago) that prove this stronger convergence, focusing on explaining the main ideas of the argument.

• March 27

  Spring Break

• April 3

  4:45 - 5:45

  Ionel Popescu (Georgia Tech)

  Stochastic Target Approach to Ricci Flow on Surfaces

  Abstract: We show that the normalized Ricci flow on surfaces of non-positive Euler characteristics converges to a metric of constant curvature. We do this using entirely probabilistic tools. First we set up a stochastic target problem on the surface and then show the $C^0$ and $C^1$ convergence. The main tool here is coupling of time changed Brownian motions. For the more critical convergence, namely $C^2$ convergence, we introduce a novel coupling of three time changed Brownian motions. This is joint work with Robert W. Neel.

• April 10

  Nina Holden (MIT)

  Percolation-decorated triangulations and their relation with SLE and LQG

  Abstract: The Schramm-Loewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We prove that a percolation-decorated RPM converges in law to SLE-decorated LQG in a certain topology. This is joint work with Bernardi and Sun. We then discuss a work in progress with the goal of strengthening the topology of convergence of RPM to LQG by considering conformal embeddings of the RPM into the complex plane. This is joint with Sun and with Gwynne, Miller, Sheffield, and Sun.

  ---

  5:20--6:20

  Jason Miller (Cambridge)

  CLE percolations

  Abstract: Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set --- a random and conformally invariant analog of the Sierpinski carpet or gasket.

  We derive a direct relationship between the conformal loop ensembles with simple loops (CLE$_\kappa$ for $\kappa \in (8/3, 4)$, whose loops are Schramm's SLE$_\kappa$-type curves) and the corresponding conformal loop ensembles with non-simple loops (CLE$_{\kappa'}$ with $\kappa' := 16/\kappa \in (4, 6)$, whose loops are SLE$_{\kappa'}$-type curves). This correspondence is the continuum analog of the {\em Edwards-Sokal} coupling between the $q$-state Potts model and the associated FK random cluster model, and its generalization to non-integer~$q$. Moreover, this is the first construction of continuum percolation on a {\em fractal} planar domain.

  Based on joint work with Scott Sheffield and Wendelin Werner.

• April 17

  Patriots Day

• April 24

  3:00 - 4:00

  Charles Smart (Cornell)

  The limit shape of convex hull peeling

  Abstract: This is joint work with Jeff Calder. Convex peeling provides a way to generalize one dimensional order statistics to higher dimensions. We prove that, under suitable conditions, the convex peeling of a random point cloud approximates the solution of a nonlinear partial differential equation. This requires identifying a suitable scale-invariant problem and using geometry to obtain tail bounds.

• May 1

  3:00 - 4:00

  Xue-Mei Li (Warwick)

  Perturbation to conservation laws

  Abstract: We study a family of singular perturbed second order differential operators.

  The unperturbed operators are invariant under group actions and so their projections to the orbit space are conservations laws, which will be used to separate the slow and the fast motions of the stochastic dynamics. At a larger scale we will see the convergence of the slow dynamics which are smooth curves to a diffusion process which we will identify.

  This study is motivated by two examples. The first is the dynamical description for a Brownian motion (on a manifold); the second is the convergence of metric spaces especially the collapsing of a family of Riemannian manifolds to a lower dimensional manifold and the convergence of their spectra.

• May 8

  Ashkan Nikeghbali (UZH)

  Some probabilistic models in arithmetic: a random matrix approach

  Abstract: We present several models for two classical arithmetic functions: the number of distinct prime divisors and the Riemann zeta function on the critical line. We try to explain to which extent these models are either successful or fail to capture the value distributions of these functions. We shall mostly focus on random matrix models and infinite dimensional objects constructed from them.

• May 15

  Alan Hammond (Berkeley)

  The weight, geometry and coalescence of scaled polymers in Brownian last passage percolation

  Abstract: In last passage percolation (LPP) models, a random environment in the two-dimensional integer lattice consisting of independent and identically distributed weights is considered. The weight of an upright path is said to be the sum of the weights encountered along the path. A principal object of study are the polymers, which are the upright paths whose weight is maximal given the two endpoints. Polymers move in straight lines over long distances with a two-thirds exponent dictating fluctuation. It is natural to seek to study collective polymer behaviour in scaled coordinates that take account of this linear behaviour and the two-third exponent-determined fluctuation.

  We study Brownian LPP, a model whose integrable properties find an attractive probabilistic expression. Building on a study arXiv:1609.02971 concerning the decay in probability for the existence of several near polymers with common endpoints, we demonstrate that the probability that there exist k disjoint polymers across a unit box in scaled coordinates has a superpolynomial decay rate in k. This result has implications for the Brownian regularity of the scaled polymer weight profile begun from rather general initial data.