CUNY Probability Seminar, Spring 2022

The CUNY Probability Seminar will have both in-person and online talks. Its usual time will be Tuesdays from 4:30 to 5:30 pm EST. The exact dates, times, and seminar links are mentioned below. If you are interested in speaking at the seminar or would like to be added or to be removed from the seminar mailing list, then please get in touch with either of the Seminar Coordinators

Matthew Junge and Emma Bailey

Seminar Schedule:

The seminar meets on Tuesdays from 4:30 to 5:30 pm EST.

The zoom link, when applicable, will be sent out via the CUNY Probability Seminar listserv. If you are not on the mailing list, please get in contact with the seminar organisers to receive the joining information.

Time: February 8, 1:30 – 2:30 pm EDT (*Different time*)
Speaker: Nicolas Curien
*Online seminar*
Title:  Around the online nearest neighbor tree
Abstract: Let points X1,X2,… be i.d.d. uniform over [0,1]d.
When a new point Xn arrives, it connects to the nearest point among X1,…,Xn-1.
This forms a sequence of trees (Tn). If you are given the sequence of unlabeled trees (Tn), can you recover information about the underlying space, in particular the dimension d?
If initially two points colored in red and blue are placed on the space, using the same procedure as above we color the new arriving points according to the color of its parent. We shall look at the interfaces between the blue points and the red points answering a few conjectures of Aldous.

Based on ongoing joint works with Jerome Casse and Alice Contat and with Anne-Laure Badesvant, Guillaume Blanc and Arvind Singh.

Time: February 15, 4:00 – 5:00 pm EDT (*Different time*)
Speaker: Emma Bailey (CUNY Math Dept Colloquium)
Location: The Graduate Center, Math Lounge 4214 (In person)
Title:  Random matrices and number theory
Abstract: This talk will present an overview of the interplay between random matrix theory and number theory, touching on connections with probability, integrable systems, and combinatorics. Originating with a conversation between Montgomery and Dyson, the apparent connections between random matrices and certain number theoretic functions — including notably the Riemann zeta function — continue to be extremely fruitful 50 years on.

Time: February 22, 1:30 – 2:30 pm EDT (*Different time*)
Speaker: Cornelia Pokalyuk
*Online seminar*
Title:  Invasion of cooperative parasites in moderately structured host populations
Abstract: Certain defense mechanisms of phages against the immune system of their bacterial host rely on cooperation of phages. Motivated by this example we analyse invasion probabilities of cooperative parasites in host populations that are moderately structured. More precisely we assume that hosts are arranged on the vertices of a configuration model and that offspring of parasites move to nearest neighbours sites to infect new hosts. We consider parasites that generate many offspring at reproduction, but do this (usually) only when infecting a host simultaneously. In this regime we identify and analyse the spatial scale of the population structure at which invasion of parasites turns from being an unlikely to an highly probable event.

Time: March 1, 10:00 – 11:00 am EDT (*Different time*)
Speaker: Perla Sousi
*Online seminar*
Title:  Phase transition for the late points of the random walk
Abstract: Let X be a simple random walk in nd with d 3 and let tcov be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set α of points that have not been visited by time αtcov and prove that it exhibits a phase transition: there exists α* so that for all α > α* and all ϵ > 0 there exists a coupling between α and two i.i.d. Bernoulli sets ± on the torus with parameters n(a±ϵ)d with the property that α+ with probability tending to 1 as n → ∞. When α α*, we prove that there is no such coupling.

Time: March 8, 4:30 – 5:30 pm EDT
Speaker: Hugo Falconet
Location: The Graduate Center, Room 9207
Title:  Metric growth dynamics in Liouville quantum gravity
Abstract: Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with conformal symmetries was introduced in the physics literature by Polyakov in the 80’s and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls. Based on a joint work with Julien Dubédat.

Time: March 15, 4:30 – 5:30 pm EDT
Speaker: Shirshendu Ganguly
Title:  Tail behaviors of non-linear observables in sparse random networks

Abstract: While the tail behavior of linear functions of i.i.d. random variables is well understood by now, in many natural settings one encounters non-linear functions of the same. Two important examples are triangle or other subgraph counts in Erdos-Renyi random graphs, where each edge in a graph of size n occurs independently with probability p, and eigenvalues of random matrices whose entries are i.i.d random variables up to symmetry constraints. A particularly central subclass of examples include sparse instances, i.e., when the edge density in a random graph goes to zero with the graph size, or analogously when only a vanishing fraction of the random matrix entries are non-zero. However, while the study of such non-linear random variables have led to spectacular progress in the theory of concentration of measure and large deviations, both for dense and sparse models, arguably the most interesting, owing to their connections to various models of statistical mechanics, sparse cases of constant average degree (e.g., Erdos-Renyi random graph of size n with p=O(1/n )), have generally remained out of reach.

Focusing on this particular sparsity regime, we will present several recent results pinning down the tail behavior of various non-linear observables, thereby settling long standing questions.

Time: March 22, 4:30 – 5:30 pm EDT
Speaker: Johannes Alt
Location: The Graduate Center, Room 9207
Title:  Localization and Delocalization in Erdős–Rényi graphs

Abstract: We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously.This is based on joint works with Raphael Ducatez and Antti Knowles.

Time: March 29, 4:30 – 5:30 pm EDT
Speaker: Jay Rosen
Location: The Graduate Center, Room 9207
Title:  Law of the iterated logarithm for k/2–permanental processes and the local times of associated Markov processes
Abstract: Let Y be a symmetric Borel right process with locally compact state space T R1 and
potential densities u(x,y) with respect to some ψ-finite measure on T. Let g and f be finite
excessive functions for Y . Set ug,f(x,y) = u(x,y) + g(x)f(y), x,y T. Let η = {η(x),x T} be a mean zero Gaussian process with covariance u(x,y) and
set ψ2(x) = E(η(x) η(0))2.
In this talk many explicit examples are given. In most of them Y is a symmetric Lévy
process or a diffusion, that is killed at the end of an independent exponential time or the first
time it hits 0.

Under general smoothness conditions on the corresponding functions g, f and ψ2, laws of
the iterated logarithm are found for Xk∕2 = {Xk∕2(t),t T}, a k∕2permanental process with kernel {ug,f(x,y),x,y T}, of the following form:
For any d T and all integers k 1,

lim sup x0(|Xk∕2(d + x) Xk∕2(d)|)(2ψ2(x) log log 1∕x)12 = (2Xk∕2(d))12, a.s.

Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the
iterated logarithm are found for the local times of Markov processes with potential densities
the same as or closely related to {ug,f(x,y),x,y T}.

Time: April 5, 4:30 – 5:30 pm EDT
Speaker: Milind Hegde
Location: The Graduate Center, Room 9207
Title: Understanding the upper tail behaviour of the KPZ equation via the tangent method

Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.

Joint work with Shirshendu Ganguly.

Time: April 12, 4:30 – 5:30 pm EDT
Speaker: Nicolas Fraiman
*Online seminar*

Title: Algorithms for the Potts model on expander graphs
Abstract: We give algorithms to sample and approximate the ferromagnetic Potts model on d-regular expander graphs. We require much weaker expansion than in previous works. The main improvements come from a significantly sharper analysis of abstract polymer models, using extremal graph theory and applications of Karger’s algorithm to counting cuts. It is #BIS-hard to approximate the partition function at low temperatures on d-regular graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. This is joint work with Charlie Carlson, Ewan Davies, Alexandra Kolla, Aditya Potukuchi and Corrine Yap.

Time: April 19, 4:30 – 5:30 pm EDT
*Spring Break: No seminar*

Time: April 26, 4:30 – 5:30 pm EDT
Speaker: Duncan Dauvergne
*Online seminar*
Title: Infection spread in a sea of random walks
Abstract: We consider a class of interacting particle systems where particles perform independent random walks on and spread an infection according to a susceptible-infected-recovered model. I will discuss a new method for understanding this model and some variants. A highlight of this method is that if recovery rate is low, then the infection survives forever with positive probability, and spreads outwards linearly leaving a herd immunity region in its wake. Based on joint work with Allan Sly.

Time: May 3, 4:30 – 5:30 pm EDT
Speaker: Chris Hoffman
Location: The Graduate Center, Room 9207
Title: Abelian Networks
Abstract: Abelian networks a class of models from statistical physics introduced as toy models to help understand the complex behavior exhibited by forest fires and avalanches. In this talk we will introduce a few different Abelian networks, give examples of the techniques used to study them and discuss some recent results.

Time: May 10, 4:30 – 5:30 pm EDT
Speaker: Asher Roberts
Location: The Graduate Center, Room 9207
Title: Selberg’s Central Limit Theorem for log|ζ(1/2+it)|
Abstract: We discuss a more recent approach to proving Selberg’s central limit theorem for the Riemann zeta function on the critical line. We will also see that it is possible to extend this approach to a multivariate context. The rate of convergence involved will come up, but there is still more work to be done here.