# USC Probability and Statistics Seminars - 2013/2014

Fridays, 3:30PM in KAP 414

Organizer: Quentin Berger ; qberger@usc.edu

For seminars/colloquia on other topics, see the Department of Mathematics webpage

• March 7th: Stephen De Salvo, UCLA
Completely effective error bounds for Stirling numbers and a Poisson limit theorem for set partitions.

We obtain explicit error bounds on an approximation for Stirling numbers of the first and second kind, $s(n,m)$ and $S(n,m)$.  These bounds imply a simple asymptotic for the Stirling numbers, in case  $n-m \sim t \sqrt{n})$ for fixed $t>0$.  The approach starts with the relation between Stirling numbers, and placement of non-challenging rooks on a triangular board, together with the Chen-Stein Theorem on Poisson approximation and a straight-forward counting argument.

Time permitting, we will also discuss a limit theorem on the sizes of blocks of a random set partition conditioned on having exactly $m$ blocks, where $m = n - t n^a$, for $t>0$, $0\leq a < 1$.

This is joint work with Richard Arratia.
• March 14th: Carl Mueller, University of Rochester

Do Stochastic PDS Hit Points in the Critical Dimension?

This talk will describe work in progress with R. Dalang, Y. Xiao, and S. Tindel. The stochastic heat equation is often used as a basic model for a moving polymer:

(1) \partial_t u = \triangle u + \dot{W} (t, x).

Here, u = u(t, x) \in \bf{R}^d is the position of the polymer, x \in \bf{R} is the length along the polymer, and t is time. \dot{W} (t, x) is two-parameter vector-valued white noise. Note that u \in R^d; that is, u is vector-valued. This is consistent with the interpretation of u as the position of a polymer. We say that the solution hits a point z if there is positive probability that u(t, x) = z for some (random) parameters (t, x). Some time ago the speaker and R. Tribe proved that d = 6 is the critical dimension for the solution u to hit points. That is, for a generic point z, the solution u hits points iff d > 6. The proof uses arguments which are specific to (1).
The goal of our work, which is still in progress, is to adapt an argument of Talagrand to study this question for equations similar to (1) but which
(1) have colored noise in place of white noise.
(2) have nonlinearities.
As usual, the critical case is by far the hardest. In fact, there are a number of results about the situation away from criticality, but they are not sharp enough to give the results we seek.

• March 28th: Gökhan Yildirim, USC
Directed polymers in a random environment with a defect line

The directed polymer in a random environment (DPRE) models a one-dimensional object interacting with disorder. The 1+1 dimensional version of the model first appeared in the physics literature as a model for the interface in two-dimensional Ising models with random exchange interaction. Since then it has been used in models of various growth phenomena: formation of magnetic domains in spin-glasses, vortex lines in superconductors, turbulence in viscous incompressible fluids (Burger turbulence), roughness of crack interfaces, and the KPZ equation. A related problem is the competition between extended and point defects as reflected in pinning phenomena, arising for example in the context of high-temperature superconductors.

We study the depinning transition of the 1+1 dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. normal potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $\beta$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $\beta$) of the annealed critical point $u_c^a=0$.

joint work with K. S. Alexander
• April 11th: Eviatar Procaccia, UCLA
Stationary aggregation processes.

Diffusion limited aggregation (DLA), was introduced by Witten and Sander over 30 years ago, as a simple model to study the geometry and dynamics of fractal physical systems.  Almost no rigorous results are known. Itai Benjamini suggested to look at aggregation models starting from an infinite line.
In this talk I'll introduce stationary versions of known aggregation models and describe some initial results.
• April 18th: David Renfrew, UCLA
• May 2nd: Samy Tindel, University of Lorraine / Joint Math Finance seminar
• PAST SEMINARS:
• September 13: Toby Johnson, University of Wahington
Stein's method and random regular graphs
• September 20: Tuan Nguyen, USC
Random covering in high dimension by a union of scaled convex sets
• September 27: Elton Hsu, Northwestern / Joint with Math Finance Colloquium
Near-Expiry Asymptotics of the Implied Volatility in Local and Stochastic Volatility Models
• October 4: Steve Kou, Columbia and NUS / Joint with Math Finance Colloquium
First Passage times of two-dimensional brownian motion
Prediction of the discovery probability of an urn sample
• October 18: Leonid Petrov, Northeastern University
Markov dynamics on interlacing arrays
• October 25: Joel Tropp, California Institute of Technology
User-friendly tail bounds for sums of random matrices
• November 1st: Richard Arratia, USC
Bounded size bias coupling: a Gamma function bound, and universal Dickman-function behavior  (joint work with Peter Baxendale)
• November 8: James Zhao, USC
Sampling Graphs with Given Degrees
• November 15: Tom Alberts, California Institute of Technology
The Gaussian Free Field, Conformal Field Theory, and Schramm-Loewner Evolution
• Saturday, December 7: Southern California Probability Symposium
Held at USC, organized by UC Santa Barbara
http://www.pstat.ucsb.edu/scps13.html
• December 13: Anirban Basak, Stanford University
Ferromagnetic Ising measures on large locally tree-like graphs.
• December 18: Zhejie Ren, École Polytechnique (WEDNESDAY, 12/18/13, KAP 245, 2:15 - 3:15 PM, joint with Math Finance)

Viscosity solution to elliptic path dependent PDEs

• Colloquium, December 18: Nizar Touzi, École Polytechnique (WEDNESDAY, 12/18/13, KAP 414, 3:30-4:30 PM)

Martingale optimal transport and martingale inequalities

• January 24: Special Colloquium, Xiangxiong Zhang, MIT (3.30pm, KAP 414)
Positivity-preserving high order accurate schemes for hyperbolic conservation laws
• January 31: Kiros Berhane, USC (Division of Biostatistics)
Bayesian mixed hidden Markov models for categorical outcomes with differential misclassification (joint work with Yue Zhang)
• February 14th: Hubert Lacoin, Université Paris Dauphine
A Mathematical Perspective on Metastable Wetting
(joint work with Augusto Teixeira)
• February 21st: Jinchi Lv, USC (Marshall School of Business)
Impacts of High Dimensionality in Finite Samples