CUNY Probability Seminar Spring 2019

The CUNY Probability Seminar is typically held on Tuesdays from 4:15 to 5:15 pm at the CUNY Graduate Center Math Department in room 5417. The exact dates, times and locations are mentioned below. If you want to talk at the seminar, or want to be added to/removed from the seminar mailing list, then please contact the Seminar Coordinator.

Elena Kosygina (https://www.baruch.cuny.edu/math/elenak/)

Seminar Schedule:

Tuesday, January 29, 2019, 4:15-5:15
Room 5417
Speaker: Jian Ding, University of Pennsylvania
Title: Random walk among Bernoulli obstacles
Abstract: Consider a discrete time simple random walk on Z^d, d\geq 2 with random Bernoulli obstacles, where the random walk will be killed when it hits an obstacle. We show that the following holds for a typical environment (for which the origin is in an infinite cluster free of obstacles): conditioned on survival up to time n, the random walk will be localized in a single island. In addition, the limiting shape of the island is a ball and the asymptotic volume is also determined. This is based on joint works with Changji Xu. Time permitting, I will also describe a recent result in the annealed case, which is a joint work with Ryoki Fukushima, Rongfeng Sun and Changji Xu.

Tuesday, February 5, 2019, 4:15-5:15
Room 5417
Speaker: Louis-Pierre Arguin, Baruch College and the CUNY Graduate Center
Title: Moments of the Riemann zeta function on a short interval
Abstract: The moments of the Riemann zeta function on the interval [T,2T] on the critical line play a fundamental role in the distribution of the prime numbers. In this talk, we look at the moments of the Riemann zeta function on typical short intervals (but with length diverging with T). We will show that the moments exhibit a freezing phase transition up to a certain interval length akin to the transition seen seen in log-correlated processes.  As a consequence we prove the leading order of the maximum of the zeta function on such short intervals. The results generalize a conjecture of Fyodorov & Keating and the related results of Arguin et al. and Najnudel on intervals of length one. Joint work with F. Ouimet and M. Radziwill.

Tuesday, February 12, 2019, 4:15-5:15  – no seminar (CUNY is closed for Lincoln’s birthday)

Tuesday, February 19, 2019, 4:15-5:15
Room 5417
Speaker:  Antonia Foldes, College of Staten Island and the CUNY Graduate Center
Title: Random walks on some planar structures
Abstract:  Link to a PDF

Tuesday, February 26, 2019, 4:15-5:15
Room 5417
Speaker: Ramon van Handel, Princeton University
Title: Extremals in Minkowski’s inequalities and degenerate diffusions
Abstract: In a seminal 1903 paper, Minkowski laid the foundation for the modern theory of convex geometry. In particular, he introduced certain fundamental inequalities that unify and extend may known inequalities such as the isoperimetric inequality, the Brunn-Minkowski inequality, etc. It has been a long-standing problem, dating back to Minkowski’s paper, to characterize the extremals in these inequalities. In this talk, I will explain how this problem in geometry can be settled by connecting it to the study of certain highly degenerate diffusions. No background in geometry will be assumed. (Joint work with Yair Shenfeld).

Tuesday, March 5, 2019, 4:15-5:15 – no seminar ( EC meeting followed by a colloquium)

Tuesday, March 12, 2019, 4:15-5:15
Room 5417
Speaker: Si Tang, Lehigh University
Title: Frog model on trees with drift
Abstract: We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a d-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. This is a joint work with E. Beckman, N. Frank, Y. Jiang, and M. Junge.

Tuesday, March 19, 2019, 4:15-5:15
Room 5417
Speaker: Alisa Knizel, Columbia University
Title: Generalization of TASEP in continuous inhomogeneous space
Abstract: We investigate a new class of exactly solvable particle systems
generalizing the Totally Asymmetric Simple Exclusion Process (TASEP).
One of the features of the particle systems we consider is the
presence of spatial inhomogeneity which can lead to the formation of
traffic jams.
For systems with special step-like initial data, we find explicit
limit shapes, describe their hydrodynamic evolution, and obtain
asymptotic fluctuation results which put our generalized TASEPs into
the Kardar-Parisi-Zhang universality class. At a critical scaling
around a traffic jam we observe deformations of the Tracy-Widom
distribution and the extended Airy kernel.

The exact solvability and asymptotic behavior of generalizations of
TASEP we study are powered by a new nontrivial connection to Schur
measures and processes.

Based on joint work with Leonid Petrov and Axel Saenz.

Tuesday, March 26, 2019, 4:15-5:15
Room 5417
Speaker: Jack Hanson, CCNY
Title: Universality of the time constant for critical first-passage percolation on the triangular lattice
Abstract: We consider first-passage percolation (FPP) on the triangular lattice with vertex weights whose common distribution function F satisfies F(0) = 1/2. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T_n the first-passage time from 0 to the boundary of the box of sidelength n, we show existence of the time constant — the limit of T_n / \log n — and find its exact value to be I / (2 \sqrt{3} pi). (Here I = \inf{x > 0 : F(x) > 1/2}.) This shows that the time constant is universal, in the sense that it is insensitive to most details of F. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of I, under the optimal moment condition on F.

Tuesday, April 2, 2019, 4:15-5:15
Room 5417
Speaker: Marcus Michelen, University of Pennsylvania
Title: Critical and Near-Critical Percolation on Galton-Watson Trees
Abstract: Letting \theta_T (p) denote the probability that the root of tree T is contained in an infinite cluster in p-percolation, we study the quenched behavior of \theta_T (p) when T is a Galton-Watson tree. For each k, a kth order Taylor expansion at criticality is obtained almost-surely, with the coefficients given as limits of tree-dependent martingales. Additionally, we construct the incipient infinite cluster on Galton-Watson trees, and prove quenched limit theorems for the size of its layers. Portions are based on joint work with Robin Pemantle and Josh Rosenberg.

Tuesday, April 9, 2019, 4:15-5:15
Room 5417
Speaker: Victor de la Pena, Columbia University
Title: The Price of Dependence is a Constant
Abstract:  In this paper I will introduce a decoupling inequality for concave functions of sums of non-negative variables which (in some sense) supports the title’s claim.

Tuesday, April 16, 2019, 4:15-5:15
Room 5417
Speaker: Firas Rassoul-Agha, University of Utah
Title: Busemann functions and Gibbs measures in directed polymer models on Z^2.
Abstract: We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity.

Tuesday, April 23, 2019  –     Spring recess – no seminar

Tuesday, April 30, 2019, 4:15-5:15
Room 5417
Speaker: Larry Goldstein, University of Southern California
Title: Dickman approximation in simulation, summations and perpetuities                      
Abstract: The generalized Dickman distribution \mathcal{D}_\theta with parameter \theta>0 is the unique solution to the distributional equality W=_d W^*, where W^*=_d U^{1/\theta}(W+1) with W non-negative with probability one, U \sim {\cal U}[0,1] independent of W, and =_d denotes equality in distribution. Members of this family appear in the study of algorithms, number theory, stochastic geometry, and perpetuities.

The Wasserstein distance d(\cdot,\cdot) between such a W with finite mean and D \sim {\cal D}_\theta obeys d(W,D) \le (1+\theta)d(W^*,W). The specialization of this bound to the case \theta=1 and coupling constructions yield for n \ge 1 that d_1(W_n,D) \le \frac{8\log (n/2)+10}{n} where W_n=\frac{1}{n}C_n-1 and C_n is the number of comparisons made by the Quickselect algorithm to find the smallest element of a list of n distinct numbers.

Joint with Bhattacharjee, using Stein’s method, bounds for Wasserstein type distances can also be computed between {\cal D}_\theta and weighted sums arising in probabilistic number theory of the form S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k \log(p_k) where (p_k)_{k \ge 1} is an enumeration of the prime numbers in increasing order and X_k is, for instance, Geometric with parameter 1-1/p_k.

Tuesday, May 7, 2019, 4:15-5:15
Room 5417
Speaker: Hanbaek Lyu, UCLA
Title: Stable network observables via dynamic embedding of motifs
Abstract: We propose a novel framework for constructing and computing various stable network observables. Our approach is based on sampling a random homomorphism from a small motif of choice into a given network. Integrals of the law of the random homomorphism induces various network observables, which include well-known quantities such as homomorphism density and average clustering coefficient. We show that these network observables are stable with respect to renormalized cut distance between networks. For their efficient computation, we also propose two Markov chain Monte Carlo algorithms and analyze their convergence and mixing times. We demonstrate how our techniques can be applied to network data analysis, especially for hypothesis testing and hierarchical clustering, through analyzing both synthetic and real world network data.

Tuesday, May 14, 2019 – no meeting