# Probability Seminar

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 617 (Temple) or David Rittenhouse Lab 4C6 (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)
TBA

James Melbourne, University of Delaware

• Tuesday March 14, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
TBA

Patrick Devlin, Rutgers

• Tuesday March 21, 2017 at 15:00, Temple (Wachmann Hall 617)
TBA
• Tuesday March 28, 2017 at 16:00, Temple (Wachman Hall 617)
TBA

Arnab Sen, University of Minnesota

• Tuesday April 4, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
TBA
• Tuesday April 18, 2017 at 15:00, UPenn (David Rittenhouse Lab 3C8)
TBA