# Autumn 2019

August 29, 2019: Matthew Wascher (OSU, Statistics)

Title: Survival dynamics for the contact process with avoidance on $Z, Z_n$, and the star graph

Abstract:

We consider the contact process with avoidance, a modified contact process, on directed graphs in which each healthy vertex can avoid each of its infected neighbors at rate $\alpha$ by turning off the directed edge from that infected neighbor to itself until the infected neighbor recovers. This model presents a challenge because, unlike the classical contact process ($\alpha = 0$,) it has not been shown to be an attractive particle system. We study the survival dynamics of this model on the lattice $Z$, the cycle $Z_n$, and the star graph. On $Z$, we show there is a phase transition in $\lambda$ between almost sure extinction and positive probability of survival. On $Z_n$, we show that as the number of vertices $n \rightarrow \infty$, there is a phase transition between log and exponential survival time in the size of the graph. On the star graph, we show that as $n \rightarrow \infty$ the survival time is polynomial in $n$ for all values of $\lambda$ and $\alpha$. This contrasts with the classical contact process where the the survival time on the star graph is exponential in $n$ for all values of $\lambda$.

September 5, 2019: Alon Nishry (Tel Aviv University)

Title:A simple construction of a completely rigid point process

Abstract:
Consider a point process in the plane with infinitely many points. For a fixed compact set K, we would like to know what the configuration of points outside of K tells us about the configuration of points inside it.

Not long ago Ghosh and Peres introduced and studied this problem for two invariant point processes, the Ginibre ensemble and zeros of the Gaussian Entire Function. I will explain some of their findings and describe a simple way to construct a Gaussian entire function whose zero set is ‘completely rigid’. This means that if the location of the zeros in the complement of a given compact set is known, then the number and location of the zeros inside that set can be determined uniquely.

Based on a joint work with A. Kiro (TAU).

September 12, 2019: Elliot Paquette

Title:The Gaussian analytic function is either bounded or covers the plane

Abstract:The Gaussian analytic function (GAF) is a power series with independent Gaussian coefficients. In the case that this power series has radius of convergence 1, it is a classical theorem that the power series is a.s. bounded on the open disk if and only if it extends continuously to a function on the closed unit disk a.s.  Nonetheless, there exists a natural range of coefficients in which the GAF has boundary values in L-p, but is a.s. unbounded.  How wild are these boundary values?  Well, Kahane asked if a GAF either a.s. extends continuously to the closed disk or a.s. has range covering the whole plane.  We discuss this and other problems and introduce a useful heuristic for understanding these problems in terms of branching processes.

September 19, 2019: Open

Title: Open

Abstract: Open

September 26, 2019: Mohamed Omar (Harvey Mudd) — Talk at 3pm, joint with GCIS seminar.

Title: Convex Intersection via Algebra

Abstract: What intersection patterns can arise from an arrangement of convex sets?  This question has received considerable attention as of late, prompted by recent discoveries in the sciences.  In this talk, we show how algebra can help us in answering this question, or at the very least unearth obstructions.

October 3, 2019: Joseph Najnudel (Bristol)

Title:  Gaussian multiplicative chaos and random matrix theory.

Abstract:  We identify an equality between two objects arising from different contexts of mathematical physics: Kahane’s Gaussian Multiplicative Chaos on the circle, and the Circular Beta Ensemble from Random Matrix Theory. This is obtained via an analysis of related random orthogonal polynomials, making the approach spectral in nature.

October 10, 2019: (Fall Break)

Title: (No seminar)

Abstract: (No seminar)

October 17, 2019: Dan Han (Louisville)

Title: Population Dynamical Systems with Immigration

Abstract: The paper builds a population dynamical system with immigration on the multidimensional lattice Z^d, d>=1 and contains several results about steady states and higher moments of population dynamics. Additional results concern the Lyapunov stability of the moments with respect to small perturbations of the parameters of the model, such as mortality rate, birth rate and the immigration rate.

October 24, 2019: Zoe Huang (Duke)

Title: The contact process on Galton-Watson trees

Abstract: The contact process describes an epidemic model where each infected individual recovers at rate 1 and infects its healthy neighbors at rate $\lambda$. We show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $\lambda_2=0$ and (ii) when it is Geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. This is based on an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs.  Recently it is proved by Bhamidi, Nam, Nguyen and Sly (2019) that when the offspring distribution of the Galton-Watson tree has exponential tail, the first critical value $\lambda_1$ of the contact process is strictly positive. We prove that if the contact process survives then the number of infected sites grows exponentially fast. As a consequence we show that the contact process dies out at the critical value $\lambda_1$ and does not survive strongly at $\lambda_2$. Based on joint work with Rick Durrett.

October 31, 2019: Yeor Hafouta (OSU)

Title:A Berry-Esseen theorem for inhomogeneous Markov chains with transition densities

Abstract: The central limit theorem (CLT) for inhomogeneous Markov chains was proved for the first time by Dobrushin in 1956, and since then there were
several extensions, under various conditions. In the talk I will discuss an optimal convergence rate (a “Berry-Esseen theorem”) in such CLT’s when the chain
is generated by certain transition densities.

The proof of Dobrushin relied on certain contraction properties of the underlying transition operators (together with Bernstein’s block method). In 2005 Sethuraman
and Varadhan proved a more general version of Dobrushin’s result using martingale approximations, and in 2012 M. Peligrad also used martingales to prove such a CLT,
but under weaker (contraction) conditions, while some other authors used mixing properties of such chains. All of these methods do not yield an optimal convergence rate,
and our results rely on relatively new theory of contraction properties of complex projective metrics. The talk is based on a joint work with Yuri Kifer.

November 7, 2019: Huseyin Acan (Drexel)

Title: Perfect matchings and Hamilton cycles in uniform attachment graphs

Abstract:  A uniform attachment graph is a random graph on the vertex set {1,…,n}, where each vertex v makes k selections from {1,…,v-1} uniformly and independently, and these selections determine the edge set. (Here k is a parameter of the graph.) The threshold k-values for the existence of a perfect matching and the existence of a Hamilton cycle are still unknown for this graph. Improving the results of Frieze, Gimenez, Pralat and  Reiniger (2019), we show that a uniform attachment graph has, with high probability, a perfect matching for k>4 and a Hamilton cycle for k> 12.

November 14, 2019: Open

Title: Open

Abstract: Open

November 21, 2019: Gaultier Lambert (Zürich)

Title:Multivariate normal approximation for traces of random unitary matrices

Abstract:

Let us consider a random matrix U of size n distributed according to the Haar measure on the unitary group. It is well-known that for any k≥1, Tr[U^k] converges as n tends to infinity to a Gaussian random variable and that, surprisingly, the speed of convergence is super exponential. In this talk, we revisit this problem and present non asymptotic bounds for the total variation distance between Tr[U^k] and a Gaussian. We will also consider the multivariate problem and explain how this affect the rate of convergence. We expect that our bounds are almost optimal. This is joint work with Kurt Johansson (KTH).

November 28, 2019: (Thanksgiving)

Title: No seminar

Abstract: No seminar

December 3, 2019 (Tuesday at 2:00 MW 154 — joint with PDE seminar): Alex Hening (Tufts)

Title: Stochastic persistence and extinction

Abstract:  A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we show how the random switching can `rescue’ species from extinction.

December 5, 2019: Open

Title: Open

Abstract: Open