# Probability Seminar 2017

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

For a chronological listing, click the year above.

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

You can also check out the seminar website at Penn.

• Tuesday January 24, 2017 at 15:00, UPenn (David Riitenhouse Lab 3C8)
Abelian squares and their progenies

Charles Burnette, Drexel University

A polynomial P ∈ C[z1, . . . , zd] is strongly Dd-stable if P has no zeroes in the closed unit polydisc D d . For such a polynomial define its spectral density function as SP (z) = P(z)P(1/z) −1 . An abelian square is a finite string of the form ww0 where w0 is a rearrangement of w. We examine a polynomial-valued operator whose spectral density function’s Fourier coefficients are all generating functions for combinatorial classes of con- strained finite strings over an alphabet of d characters. These classes generalize the notion of an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, L2 (T d)-valued operator. Integral representations and asymptotic behavior of the coefficients of these generating functions and a combinatorial meaning to Parseval’s equation are given as consequences.

• Tuesday January 31, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
How round are the complementary components of planar Brownian motion?

Nina Holden, MIT

Consider a Brownian motion $W$ in the complex plane started from $0$ and run for time $1$. Let $A(1), A(2),...$ denote the bounded connected components of $C-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i \in N$. Our main result is that $E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty$ for any $\theta <1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

• Tuesday February 7, 2017 at 15:00, Temple (Wachmann Hall 617)
Stochastic areas and Hopf fibrations

Fabrice Baudoin, University of Connecticut

We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces ℂℙn and ℂℍn. The characteristic functions of those processes are computed and limit theorems are obtained. For ℂℙn the geometry of the Hopf fibration plays a central role, whereas for ℂℍn it is the anti-de Sitter fibration. This is joint work with Jing Wang (UIUC).

• Tuesday February 14, 2017 at 15:00, Temple (Wachmann Hall 617)
Intermediate disorder limits for multi-layer random polymers

Mihai Nica, NYU

The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble. Part of this talk is based on joint work with I. Corwin.

• Tuesday February 21, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
Large deviation and counting problems in sparse settings

Shirshendu Ganguly, Berkeley

The upper tail problem in the Erdös-Rényi random graph $G \sim G(n,p)$, where every edge is included independently with probability $p$, is to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1 + d$. The arithmetic analog considers the count of arithmetic progressions in a random subset of $Z/nZ$, where every element is included independently with probability $p$. In this talk, I will describe some recent results regarding the solution of the upper tail problem in the sparse setting i.e. where $p$ decays to zero, as $n$ grows to infinity. The solution relies on non-linear large deviation principles developed by Chatterjee and Dembo and more recently by Eldan and solutions to various extremal problems in additive combinatorics.

• Tuesday February 28, 2017 at 15:00, Temple (Wachmann Hall 617)
Bounds on the maximum of the density for certain linear images of independent random variables

James Melbourne, University of Delaware

We will present a generalization of a theorem of Rogozin that identifies uniform distributions as extremizers of a class of inequalities, and show how the result can reduce specific random variables questions to geometric ones. In particular, by extending "cube slicing" results of K. Ball, we achieve a unification and sharpening of recent bounds on densities achieved as projections of product measures due to Rudelson and Vershynin, and the bounds on sums of independent random variable due to Bobkov and Chistyakov. Time permitting we will also discuss connections with generalizations of the entropy power inequality.

• Tuesday March 21, 2017 at 15:00, Temple (Wachmann Hall 617)
Local extrema of random matrices and the Riemann zeta function

Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to conjecture the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on a joint works with Arguin, Belius, Radziwill and Soundararajan.

• Tuesday March 28, 2017 at 15:00, Temple (Wachman Hall 617)
Majority dynamics on the infinite 3-regular tree

Arnab Sen, University of Minnesota

The majority dynamics on the infinite 3-regular tree can be described as follows. Each vertex of the tree has an i.i.d. Poisson clock attached to it, and when the clock of a vertex rings, the vertex looks at the spins of its three neighbors and flips its spin, if necessary, to come into agreement with majority of its neighbors. The initial spins of the vertices are taken to be i.i.d. Bernoulli random variables with parameter p. In this talk, we will discuss a couple of new results regarding this model. In particular, we will show that the limiting proportion of ‘plus’ spins in the tree is continuous with respect to the initial bias p. A key tool in our argument is the mass transport principle. The talk is based on an ongoing work with M. Damron.

• Tuesday April 4, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
Galton-Watson fixed points, tree automata, and interpretations

Tobias Johnson, NYU

onsider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

• Tuesday April 11, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
Biased random permutations are predictable (proof of an entropy conjecture of Leighton and Moitra)

Patrick Devlin, Rutgers

Suppose F is a random (not necessarily uniform) permutation of {1, 2, ... , n} such that |Prob(F(i) < F(j)) -1/2| > epsilon for all i,j. We show that under this assumption, the entropy of F is at most (1-delta)log(n!), for some fixed delta depending only on epsilon [proving a conjecture of Leighton and Moitra]. In other words, if (for every distinct i,j) our random permutation either noticeably prefers F(i) < F(j) or prefers F(i) > F(j), then the distribution inherently carries significantly less uncertainty (or information) than the uniform distribution.

Our proof relies on a version of the regularity lemma, a combinatorial bookkeeping gadget, and a few basic probabilistic ideas. The talk should be accessible for any background, and we will gently recall any relevant notions (e.g., entropy) as needed. Those unhappy with the talk are welcome to form an unruly mob to depose the speaker, and pitchforks and torches will be available for purchase.

This is from a recent paper joint with Huseyin Acan and Jeff Kahn.

• Tuesday April 25, 2017 at 15:00, Temple (Wachman Hall 617)

Christopher Sinclair, University of Oregon

We consider the distribution of N p-adic particles with interaction energy proportional to the log of the p-adic distance between two particles. When the particles are constrained to the ring of integers of a local field, the distribution of particles is proportional to a power of the p-adic absolute value of the Vandermonde determinant. This leads to a first question: What is the normalization constant necessary to make this a probability measure? This sounds like a triviality, but this normalization constant as a function of extrinsic variables (like number of particles, or temperature) holds much information about the statistics of the particles. Viewed another way, this normalization constant is a p-adic analog of the now famous Selberg integral. While a closed form for this seems out of reach, I will derive a remarkable identity that may hold the key to unlocking more nuanced information about p-adic electrostatics. Along the way we’ll find an identity for the generating function of probabilities that a degree N polynomial with p-adic integer coefficients split completely. Joint work with Jeff Vaaler.

• Tuesday May 2, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
Percolation in Weighted Random Connection Model

When modeling the spread of infectious diseases, it is important to incorporate risk behavior of individuals in a considered population. Not only risk behavior, but also the network structure created by the relationships among these individuals as well as the dynamical rules that convey the spread of the disease are the key elements in predicting and better understanding the spread. We propose the weighted random connection model, where each individual of the population is characterized by two parameters: its position and risk behavior. A goal is to model the effect that the probability of transmissions among individuals increases in the individual riskfactors, and decays in their Euclidean distance. Moreover, the model incorporates a combined risk behavior function for every pair of theindividuals, through which the spread can be directly modeled or controlled. The main results are conditions for the almost sure existence of an infinite cluster in the weighted random connection model. We use results on the random connection model and sitepercolation in Z^2.

• Tuesday September 5, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)
Large deviations for first passage percolation

Allan Sly, Princeton

We establish a large deviation rate function for the upper tail of first passage percolation answering a question of Kesten who established the lower tail in 1986. Moreover, conditional on the large deviation event, we show that the minimal cost path is delocalized, that is it moves linearly far from the straight line path. Joint work with Riddhipratim Basu (Stanford/ICTS) and Shirshendu Ganguly (UC Berkeley).

• Tuesday September 12, 2017 at 15:00, Temple (Wachman Hall 617)
Stability of phases and interacting particle systems

Nick Crawford, Technion

I will discuss recent work with W. de Roeck on the following natural question: Given an interacting particle system are the stationary measures of the dynamics stable to small (extensive) perturbations? In general, there is no reason to believe this is so and one must restrict the class of models under consideration in one way or another. As such, I will focus in this talk on the simplest setting for which one might hope to have a rigorous result: attractive Markov dynamics (without conservation laws) relaxing at an exponential rate to its unique stationary measure in infinite volume. In this case we answer the question affirmatively.

As a consequence we show that ferromagnetic Ising Glauber dynamics is stable to small, non-equilibrium perturbations in the entire uniqueness phase of the inverse temperature/external field plane. It is worth highlighting that this application requires new results on the (exponential) rate of relaxation for Glauber dynamics defined with non-zero external field.

• Tuesday September 19, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)
Invasion percolation on Galton-Watson trees

Marcus Michelin, UPenn

Given an infinite rooted tree, how might one sample, nearly uniformly, from the set of paths from the root to infinity? A number of methods have been studied including homesick random walks, or determining the growth rate of the number of self-avoiding paths. Another approach is to use percolation. The model of invasion percolation almost surely induces a measure on such paths in Galton-Watson trees, and we prove that this measure is absolutely continuous with respect to the limit uniform measure as well as other properties of invasion percolation. This work in progress is joint with Robin Pemantle and Josh Rosenberg.

• Tuesday September 26, 2017 at 16:00, UPenn (David Rittenhouse Lab 4C8)
Cutoff for random to random

Evita Nestoridi, Princeton

Random to random is a card shuffling model that was created to study strong stationary times. Although the mixing time of random to random has been known to be of order n log n since 2002, cutoff had been an open question for many years, and a strong stationary time giving the correct order for the mixing time is still not known. In joint work with Megan Bernstein, we use the eigenvalues of the random to random card shuffling to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n log n$, answering a conjecture of Diaconis.

• Tuesday October 3, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)
Lattice path enumeration, multivariate singularity analysis, and probability theory

Stephen Melczar, UPenn

The problem of enumerating lattice paths with a fixed set of allowable steps and restricted endpoint has a long history dating back at least to the 19th century. For several reasons, much research on this topic over the last decade has focused on two dimensional lattice walks restricted to the first quadrant, whose allowable steps are "small" (that is, each step has coordinates +/- 1, or 0). In this talk we relax some of these conditions and discuss recent work on walks in higher dimensions, with non-small steps, or with weighted steps. Particular attention will be given to the asymptotic enumeration of such walks using representations of the generating functions as diagonals of rational functions, through the theory of analytic combinatorics in several variables. Several techniques from computational and experimental mathematics will be highlighted, and open conjectures of a probabilistic nature will be discussed.

• Tuesday October 10, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)
Rigidity of the 3D hierarchical Coulomb gas

Sourav Chatterjee, Stanford

The mathematical analysis of Coulomb gases, especially in dimensions higher than one, has been the focus of much recent activity. For the 3D Coulomb, there is a famous prediction of Jancovici, Lebowitz and Manificat that if N is the number of particles falling in a given region, then N has fluctuations of order cube-root of E(N). I will talk about the recent proof of this conjecture for a closely related model, known as the 3D hierarchical Coulomb gas. I will also try to explain, through some toy examples, why such unusually small fluctuations may be expected to appear in interacting gases.

• Tuesday October 17, 2017 at 15:00, Temple (Wachman Hall 617)
Homogenization of a class of 1-D nonconvex viscous Hamilton-Jacobi equations with random potential

Atilla Yilmaz, NYU & Koc University

There are general homogenization results in all dimensions for (inviscid and viscous) Hamilton-Jacobi equations with random Hamiltonians that are convex in the gradient variable. Removing the convexity assumption has proved to be challenging. There was no progress in this direction until two years ago when the 1-D inviscid case was settled positively and several classes of (mostly inviscid) examples for which homogenization holds were constructed as well as a 2-D inviscid counterexample. Methods that were used in the inviscid case are not applicable to the viscous case due to the presence of the diffusion term.

In this talk, I will present a new class of 1-D viscous Hamilton-Jacobi equations with nonconvex Hamiltonians for which homogenization holds. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial data have representations involving exponential expectations of controlled Brownian motion in random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in random potential. The proof relies on (i) analyzing the large deviation behavior of the controlled Brownian particle which assumes the role of one of the players in an emergent two-player game, (ii) identifying asymptotically optimal control policies and (iii) constructing correctors which lead to exponential martingales.

Based on recent joint work with Elena Kosygina and Ofer Zeitouni.

• Tuesday October 24, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)
Extreme level sets of branching Brownian motion

Lisa Hartung, NYU

We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida. (Joint work with A. Cortines, O Louidor.)

• Tuesday November 7, 2017 at 15:00, 617 Wachman Hall
Spectrum of random band matrices

Indrajit Jana, Temple University

We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.

• Tuesday November 14, 2017 at 15:00, UPenn (David Rittenhouse Lab 4C8)