Spring 2017
Monday 4.15  5.15 pm
Room 4153
Schedule

February 13
NO SEMINAR (snow)

February 21
TUESDAY, 4:15  5:15, ROOM 4257
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 nonintegrable discrete system: graph homomorphisms from Z^d to a regular tree. We will also explain how the technique used could be applied to other nonintegrable 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, DuminilCopin, 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 WeilPetersson metric

March 7
TUESDAY, 4:155:15, ROOM 2361
VictorEmmanuel 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 meanweight cycles
Abstract: We consider the minimum meanweight cycle (MMWC) in the stochastic meanfield 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 looperased walk in the natural parametrization
Abstract: Looperased random walk (LERW) is the random selfavoiding 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 socalled natural parametrization, which in this case is the same as 5/4dimensional 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 nonpositive 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)
Percolationdecorated triangulations and their relation with SLE and LQG
Abstract: The SchrammLoewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of twodimensional 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 percolationdecorated RPM converges in law to SLEdecorated 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.
5:206:20
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 nonsimple 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 EdwardsSokal} coupling between the $q$state Potts model and the associated FK random cluster model, and its generalization to noninteger~$q$. 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 scaleinvariant problem and using geometry to obtain tail bounds.

May 1
3:00  4:00
XueMei 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 twodimensional 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 twothirds 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 twothird exponentdetermined 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.
Spring 2017 Organizers
 Stéphane Benoist
 Alexei Borodin
 Alexey Bufetov
 Vadim Gorin
 Boris Hanin
 Elchanan Mossel
 Philippe Rigollet
 Scott Sheffield