# Probability Seminar 2016

The seminar is jointly organized between Temple and Penn, by Brian Rider (Temple) and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 pm and are held either in Wachman 617 (Temple) or David Rittenhouse Lab 4C6 (Penn).

You can also check out the seminar website at Penn.

• Tuesday January 26, 2016 at 14:30, David Rittenhouse Lab 3C8

• Tuesday February 2, 2016 at 14:30, David Rittenhouse Lab 3C8

Eric Slivken, UC Davis

• Tuesday February 9, 2016 at 14:30, David Rittenhouse Lab 3C8

Nayantara Bhatnagar, Univ Delaware

• Tuesday February 16, 2016 at 14:30, Wachman 617

Yuri Bakhtin, NYU

• Tuesday March 1, 2016 at 14:30, David Rittenhouse Lab 3C8

Sumit Mukherjee, Columbia

• Tuesday March 15, 2016 at 14:30, Wachman 617

Philippe Sosoe, Harvard

• Tuesday March 22, 2016 at 14:30, Wachman 617

Christian Benes, CUNY

• Tuesday March 29, 2016 at 14:30, David Rittenhouse Lab 3C8

John Pike, Cornell

• Tuesday April 5, 2016 at 14:30, David Rittenhouse Lab 3C8
Nodal sets of random eigenfunctions of the harmonic oscillator

Boris Hanin, MIT

• Tuesday April 12, 2016 at 14:30, David Rittenhouse Lab 3C8
Stochastic approach to anomalous diffusion in two dimensional, incompressible, periodic, celluar flows

Zsolt Pajor-Gyulai, Courant

• Tuesday April 19, 2016 at 14:30, David Rittenhouse Lab 3C8
Markov chain convergence via regeneration

Dan Jerison, Cornell

• Tuesday April 26, 2016 at 14:30, David Rittenhouse Lab 3C8
The frog model with drift on R

Josh Rosenberg, Penn

• Tuesday September 6, 2016 at 15:00, UPenn (DRL 4C6)
Random planar metrics of Gaussian free fields

Jian Ding, Chicago

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.

• Tuesday September 13, 2016 at 15:00, UPenn (DRL 4C6)
An Introduction to Limit Theorems for Nonconventional Sums

Yuri Kifer, Hebrew University

I'll survey some of the series results on limit theorems for nonconventional sums of the form $\sum_{n=1}^NF(X_n,X_{2n},...,X_{\ell n})$ and more general ones, where $\{ X_n\}$ is a sequence of random variables with sufficiently weak dependence.

• Tuesday September 20, 2016 at 15:00, Temple (Wachman 617)
Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension

Wei Wu, NYU

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and $\Phi^4$ models for $d \geq 4$. We describe a simple spin model from uniform spanning forests in $\mathbb{Z}^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension $4$, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in $d=4$, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

• Tuesday September 27, 2016 at 15:00, UPenn (DRL 4C6)
Corners in tree-like tableaux

Amanda Lohss, Drexel

Tree–like tableaux are combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. We have proven both conjectures based on a bijection with permutation tableaux and type–B permutation tableaux. In addition, we have shown that the number of diagonal boxes in symmetric tree–like tableaux is asymptotically normal and that the number of occupied corners in a random tree–like tableau is asymptotically Poisson. This extends earlier results of Aval, Boussicault, Nadeau, and Laborde Zubieta, respectively.

• Tuesday October 4, 2016 at 15:00, Temple (Wachman 617)
Chaining, interpolation, and convexity

Ramon van Handel, Princeton

A significant achievement of modern probability theory is the development of sharp connections between the boundedness of random processes and the geometry of the underlying index set. In particular, the generic chaining method of Talagrand provides in principle a sharp understanding of the suprema of Gaussian processes. The multiscale geometric structure that arises in this method is however notoriously difficult to control in any given situation. In this talk, I will exhibit a surprisingly simple but very general geometric construction, inspired by real interpolation of Banach spaces, that is readily amenable to explicit computations and that explains the behavior of Gaussian processes in various interesting situations where classical entropy methods are known to fail.

• Tuesday October 11, 2016 at 15:00, UPenn (DRL 4C6)
The front location for branching Brownian motion with decay of mass

I will describe joint work with Sarah Penington (Oxford). Consider a standard branching Brownian motion whose particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle $x$, then the mass of particle $x$ decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles. One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM, and martingales are hard to come by. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance $\sim c t^{1/3}$ behind the typical BBM front. At a high level, our argument for this may be described as a proof by contradiction combined with fine estimates on the probability Brownian motion stays in a narrow tube of varying width.

• Tuesday October 18, 2016 at 15:00, UPenn (DRL 4C6)
Random discrete structures: Scaling limits and universality

Sanchayan Sen, Eindhoven

One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent $\tau>3$, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}$. In other words, the degree exponent determines the universality class the random graph belongs to. More generally, recent research has provided strong evidence to believe that several objects constructed on a wide class of random discrete structures including (a) components under critical percolation, (b) the vacant set left by a random walk, and (c) the minimal spanning tree, viewed as metric measure spaces converge, after scaling the graph distance, to some random fractals, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures. Based on joint work with Shankar Bhamidi, Nicolas Broutin, Remco van der Hofstad, and Xuan Wang.

• Tuesday October 25, 2016 at 15:00, UPenn (DRL 4C6)
Asymptotics of stochastic particle systems via Schur generating functions

Alexey Bufetov, MIT

We will discuss a new approach to the analysis of the global behavior of stochastic discrete particle systems. This approach links the asymptotics of these systems with properties of certain observables related to the Schur symmetric functions. As applications of this method, we prove the Law of Large Numbers and the Central Limit Theorem for various models of random lozenge and domino tilings, non-intersecting random walks, and decompositions of tensor products of representations of unitary groups. Based on joint works with V. Gorin and A. Knizel.

• Tuesday November 1, 2016 at 15:00, UPenn (DRL 4C6)
Markov Chains of Exchangeable Structures

Henry Towsner, UPenn

The Aldous-Hoover Theorem characterizes arrays of random variables which are exchangeable - that is, the distribution is invariant under permutations of the indices of the array. We consider the extension to exchangeable Markov chains. In order to give a satisfactory classification, we need an extension of the Adous-Hoover Theorem to "relatively exchangeable" arrays, which are only invariant under some permutations. Different families of permutations lead to different characterization theorems, with the crucial distinction coming from a model theoretic characterization of the way finite arrays can be amalgamated.

• Tuesday November 8, 2016 at 15:00, UPenn (DRL 4C6)
Local max-cut in smoothed polynomial time

Sébastien Bubek, Microsoft

The local max-cut problem asks to find a partition of the vertices in a weighted graph such that the cut weight cannot be improved by moving a single vertex (that is the partition is locally optimal). This comes up naturally, for example, in computing Nash equilibrium for the party affiliation game. It is well-known that the natural local search algorithm for this problem might take exponential time to reach a locally optimal solution. We show that adding a little bit of noise to the weights tames this exponential into a polynomial. In particular we show that local max-cut is in smoothed polynomial time (this improves the recent quasi-polynomial result of Etscheid and Roglin). Joint work with Omer Angel, Yuval Peres, and Fan Wei.

• Tuesday November 15, 2016 at 15:00, Temple (Wachman 617)
The law of fractional logarithm in the GUE minor process

Elliot Paquette, Ohio State

Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry. The eigenvalues of the upper-left NxN submatrices, form what is called the GUE minor process. This largest-eigenvalue process is a canonical example of the Airy process which is connected to many other growth processes. We show that if one lets N vary over all natural numbers, then the sequence of largest eigenvalues satisfies a 'law of fractional logarithm,' in analogy with the classical law of iterated logarithm for simple random walk. This GUE minor process is determinantal, and our proof relies on this. However, we reduce the problem to correlation and decorrelation estimates that must be made about the largest eigenvalues of pairs of GUE matrices, which we hope is useful for other similar problems.

This is joint work with Ofer Zeitouni.

• Tuesday November 29, 2016 at 15:00, Temple (Wachman 617)
Arm events in invasion percolation

Jack Hanson, CUNY

Invasion percolation is a "self-organized critical" distribution on random subgraphs of Z^2, believed to exhibit much of the same behavior as critical percolation models. Self-organization means that this happens spontaneously without tuning some parameter to a critical value. In two dimensions, some aspects of the invasion graph are known to correspond to those in critical models, and some differences are known. We will discuss new results on the probabilities of various "arm events" -- events that connections from the origin to a large distance n are either present or "closed" in the invasion graph. We show that some of these events have probabilities obeying power laws with the same power as in the critical model, while all others differ from the critical model's by a power of n.