## Seminar Schedule:

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

**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 X

_{1},X

_{2},… be i.d.d. uniform over [0,1]

^{d}.

When a new point X

_{n}arrives, it connects to the nearest point among X

_{1},…,X

_{n-1}.

This forms a sequence of trees (T

_{n}). If you are given the sequence of unlabeled trees (T

_{n}), 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 ℤ

_{n}

^{d}with d ≥ 3 and let t

_{cov}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 ℒ

_{α}of points that have not been visited by time αt

_{cov}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 ℒ

_{α}and two i.i.d. Bernoulli sets ℬ

^{±}on the torus with parameters n

^{–(a±ϵ)d}with the property that ℬ

^{–}⊂ ℒ

_{α}⊂ ℬ

^{+}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 ⊆ R

^{1}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 u

_{g,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 ψ

^{2}(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 ψ^{2}, laws of

the iterated logarithm are found for X_{k∕2} = {X_{k∕2}(t),t ∈ T}, a k∕2–permanental process with kernel {u_{g,f}(x,y),x,y ∈ T}, of the following form:

For any d ∈ T and all integers k ≥ 1,

lim sup _{x↓0}(|X_{k∕2}(d + x) – X_{k∕2}(d)|)(2ψ^{2}(x) log log 1∕x)^{–1∕2} = (2X_{k∕2}(d))^{1∕2}, 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 {u_{g,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.