## Probability seminar

• Title: COMPARING RETURN PROBABILITIES
• Speaker: Russell Lyons (Indiana University)
• Time: 2:30 p.m., Monday, December 5, 2016
• Room: THO 325
• Abstract: Consider a Cayley graph of a group, $\Gamma$. Suppose that $W$ is a random assignment of nonnegative numbers to the edges and that the law of $W$ is $\Gamma$-invariant. Let $X_t$ be continuous-time random walk on $\Gamma$ in the random environment $W$: incident edges $e$ are crossed at rate $W(e)$. Write $p^W(t) := {\bf E}\bigl[{\bf P}_o[X_t = o]\bigr]$ for the expected return probability at time $t$ (averaged over $W$). Fontes and Mathieu asked whether given two such environments, $W_1$ and $W_2$, with $W_1(e) \le W_2(e)$ for all edges $e$, one has $p^{W_1}(t) \ge p^{W_2}(t)$ for all $t \ge 0$. When the pair $(W_1, W_2)$ has a $\Gamma$-invariant law, this was shown by Aldous and the speaker. It remains open in general. We attempt to attack this problem via similar questions for finite graphs.

• Title: SUB-CRITICAL STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
• Speaker: Zhenan Wang (University of Washington)
• Time: 2:30 p.m., Monday, November 28, 2016
• Room: THO 325
• Abstract: We will start with the notion of criticality in stochastic partial differential equations of the following type. $$u_t=div(A\nabla u)+div F+f+(G_i\cdot\nabla u+g)\dot{w}_t^i$$ We will then introduce regularity results for SPDEs with sub-critical perturbation terms. Among these sub-critical terms, we will specifically talk about properties of the random drift terms, $G_i\cdot\nabla u$, and develop techniques to show the regularity property of the solutions. A few problems regarding the critical terms will also be introduced.

• Title: CONTROLLED DIFFUSION ON GRAPHS
• Speaker: Miklos Racz (Microsoft Research)
• Time: 2:30 p.m., Monday, November 21, 2016
• Room: THO 325
• Abstract: Starting from a unit mass on a vertex of a graph, we investigate the minimum number of "controlled diffusion" steps needed to transport a constant mass outside of the ball of radius $n$. In a step of a controlled diffusion process we may select any site with positive mass and topple its mass equally to its neighbors. Our main result shows that on $Z^d$, on the order of $n^{d+2}$ steps are necessary and sufficient. We also present sharp bounds for several other graphs, such as the comb, regular trees, Galton-Watson trees, and more. This is joint work with Laura Florescu and Yuval Peres.

• Title: TWO CLUSTERING PROBLEMS FOR THE STOCHASTIC BLOCK MODEL
• Speaker: Ioana Dumitriu (University of Washington)
• Time: 2:30 p.m., Monday, November 14, 2016
• Room: THO 325
• Abstract: The Stochastic Block Model (SBM) and its variants are the most widely-used "synthetic" examples for network modeling. They are used for devising, testing, analyzing and benchmarking algorithms in community detection. The problem is formulated as follows: given a network (perhaps known through its local connections, i.e., the adjacency matrix) created by sampling the SBM, under what parameter regimes is it possible to devise an algorithm that correctly or approximately identifies the "clusters" or "communities" in the network? Any such algorithm would of course only work with high probability, as the model could in principle generate the empty graph or the complete graph, which have no non-trivial communities. The analysis that is needed in order to calculate or estimate is quite complex and it involves non-trivial random matrix theory results, as well as combinatorics, convex optimization, linear algebra, etc. While a lot has been learned in the last five years about the regimes in which such algorithms may exist, the problem in its generality is still wide open. We will discuss progress we made on a couple of models and talk about a number of open questions.

• Title: COMPACTNESS AND LARGE DEVIATIONS
• Speaker: Chiranjib Mukherjee (Courant Institute of Mathematical Sciences)
• Time: 2:30 p.m., Monday, October 31, 2016
• Room: THO 325
• Abstract: In a reasonable topological space, large deviation estimates essentially deal with probabilities of events that are asymptotically (exponentially) small, and in a certain sense, quantify the rate of these decaying probabilities. In such estimates, upper bounds for such small probabilities often require compactness of the ambient space, which is often absent in problems arising in statistical mechanics (for example, distributions of local times of Brownian motion in the full space $\mathbb R^d$).
Motivated by a problem in statistical mechanics, we present a robust theory of translation-invariant compactification" of probability measures in $\mathbb R^d$. Thanks to an inherent shift-invariance of the underlying problem, we are able to apply this abstract theory painlessly and solve a long standing problem in statistical mechanics, the mean-field polaron problem.
This talk is based on joint works with S. R. S. Varadhan (New York), as well as with Erwin Bolthausen (Zurich) and Wolfgang Koenig (Berlin).

• Title: A LAW OF THE ITERATED LOGARITHM FOR GRENANDER'S ESTIMATOR
• Speaker: Jon Wellner (University of Washington)
• Time: 2:30 p.m., Monday, October 24, 2016
• Room: THO 325
• Abstract: I will discuss the following law of the iterated logarithm for the Grenander estimator (or MLE) $\widehat{f}_n$ of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) < 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \begin{eqnarray*} \phantom{bla} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where $$M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3} \ \ \ \mbox{and} \ \ \ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} ;$$ here ${\cal G}$ is the two-sided Strassen limit set on $\mathbb{R}$. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion. The constant in the limsup is related to the tail behavior of Chernoff's distribution, the law of $\mbox{argmax} \{W(u) - u^2\}$ where $W$ is two-sided Brownian motion starting at $0$. I will also briefly discuss several open problems connected to the asymptotic distribution of the mode estimator $M(\widehat{f}_n)$ where $\widehat{f}_n$ is the MLE of a log-concave density $f$ on $\mathbb{R}$ with $(-\log f)^{\prime \prime } (m) > 0$.

• Title: BOUNDARY TRACE OF A CLASS OF DIFFUSION PROCESSES
• Speaker: Lidan Wang (University of Washington)
• Time: 2:30 p.m., Monday, October 17, 2016
• Room: THO 325
• Abstract: For a reflecting Bessel process, the inverse local time at 0 is an $\alpha$-stable subordinator, then the corresponding subordinate Brownian motion is a symmetric $2\alpha$-stable process. Based on a discussion of Esscher and Girsanov transforms of general diffusions, we would get a comparison theorem between inverse local times of Bessel processes and perturbed Bessel processes. An immediate application would be Green function estimates of trace processes.
This is joint work with Prof. Zhen-Qing Chen.

• Title: LOCAL UNIVERSALITY OF ROOTS OF RANDOM POLYNOMIALS
• Speaker: Oanh Nguyen (Yale University)
• Time: 2:30 p.m., Monday, October 10, 2016
• Room: THO 325
• Abstract: We consider random polynomials of the form $$P_n(x) = \xi_1 p_1(x) + \xi_2 p_2(x) + \dots +\xi_n p_n(x)$$ where $\xi_1, \dots, \xi_n$ are independent random variables and $p_1, \dots, p_n$ are deterministic polynomials. Questions about the distribution of the zeros of $P_n$ have attracted intensive research for many decades with seminal papers by Kac, Littlewood-Offord, Erdos-Offord, and Tao-Vu, to name a few. In this talk, we will discuss some universality properties of the roots of generalized Kac polynomials and trigonometric random polynomials. As an application, we calculate the average number of real roots and discuss some asymptotic behavior of this number. The talk is based on some joint works with Yen Do, Hoi Nguyen, and Van Vu.

• Title: INFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES
• Speaker: Andrey Sarantsev (University of California, Santa Barbara)
• Time: 2:30 p.m., Monday, October 3, 2016
• Room: THO 325
• Abstract: We consider infinite systems (one- or two-sided) of rank-based particles on the real line. We find stable distributions and convergence results for the gaps between particles.