Because of the COVID-19 outbreak, the CUNY Probability Seminar will be held by videoconference for the rest of the semester. Its usual time will be Tuesdays from 3:30 to 4:30. 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.

*Tobias Johnson (https://www.math.csi.cuny.edu/~tobiasljohnson*)

## Seminar Schedule:

**Tuesday, April 28, 2020, 3:30-4:30**

**Videoconference Link:** Zoom

**Speaker:** Jacob Richey, University of Washington

**Title: **Recent results on the phase transition for activated random walk

**Abstract: **In this talk I will discuss activated random walk, an interacting particle system that exhibits a phase transition on infinite domains and self-organized criticality on finite domains. In the infinite version, the system is initialized with density μ of particles which perform independent simple random walk, fall asleep at rate λ, and are woken up if another particle moves to the same site. There are two possible limiting behaviors: local fixation, where each site is visited finitely many times a.s., or non-fixation, where each site is visited infinitely many times a.s. Current research is focused on determining where the transition between these two phenomena occurs in terms of μ and λ, and many questions still remain – even on Z. I will present some recent results and discuss the novel tools involved. And here are the slides that will be used in the talk.

## Previous Talks

**Tuesday, February 4, 2020, 4:15-5:15**

**Room 5417**

**Speaker: Eliran Subag, Courant Institute**

**Title: **Geometric TAP approach for spherical spin glasses

**Abstract: **The celebrated Thouless-Anderson-Palmer approach suggests that the free energy of a mean-field spin glass model is achieved by maximizing a certain free energy landscape over the space of all possible magnetization vectors. I will describe a new geometric approach to define such free energy landscapes for general spherical mixed p-spin models and derive from them a generalized TAP representation for the free energy. I will then explain how these landscapes are related to the so-called pure states decomposition, ultrametricity property, and optimization of full-RSB models.

**Tuesday, February 11, 2020, 4:15-5:15**

**Room 5417**

**Speaker: Peter Winkler, Dartmouth University**

**Title: ** Permutation Pattern Densities

**Abstract: **The “pattern density” of a permutation π in a permutation σ is the fraction of subsequences of σ (written in one-line form) that are ordered like π. For example, the density of the pattern “12” in σ is the number of pairs i < j with σ(i) < σ(j), divided by .

What does a typical permutation look like that has one or more pattern densities fixed? To help answer this we employ limit objects called "permutons," together with a variational principle that identifies the permuton that best represents a given class of permutations.

Joint work with Rick Kenyon, Dan Král’ and Charles Radin, and (later) with Chris Coscia and Martin Tassy.

**Tuesday, March 3, 2020, 4:15-5:15**

**Room 5417**

**Speaker: Danny Nam, Princeton University**

**Title: **The contact process on random graphs and Galton-Watson trees

**Abstract: **The contact process describes an elementary epidemic model, where each infected vertex gets healed at rate 1 while it passes its disease to each of its neighbors independently at rate λ. On the infinite d-regular tree with the initial infection at its root, [Pemantle ’92] proved that the process has three different phases depending on λ: extinction, weak survival, and strong survival. In this talk, we show that the phase diagram of the contact process on a Galton-Watson tree depends on the tail decay of the offspring distribution, answering a question of Durrett. When the threshold is positive, we further obtain the first-order asymptotics of its location. Analogous results for Erdős-Rényi and other random graphs will be discussed as well.

Joint work with Shankar Bhamidi, Oanh Nguyen and Allan Sly

**Tuesday, March 10, 2020, 4:15-5:15**

**Room 5417**

**Speaker: Brian Rider, Temple University**

**Title: **A general beta crossover ensemble

**Abstract: **I’ll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a cross-over between the more well-known “soft” and “hard” edge point processes. Our new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. This suggests a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at β = 2. Joint work with Jose Ramírez (Univ. Costa Rica).

**Tuesday, March 24, 2020, 3:30-4:30**

**Videoconference Link:** Zoom

**Speaker: **Louis-Pierre Arguin, Baruch

**Title: **Large Values of the Riemann Zeta Function in Short Intervals

**Abstract: **In a seminal paper in 2012, Fyodorov & Keating proposed a series of conjectures describing the statistics of large values of zeta in short intervals of the critical line. In particular, they relate these statistics to the ones of log-correlated Gaussian fields. In this lecture, I will present recent results that answer many aspects of these conjectures. Connections to problems in number theory will also be discussed.

**Tuesday, March 31, 2020, 3:30-4:30**

**Videoconference Link:** Zoom

**Speaker:** Jacopo Borga, ETH-Zurich

**Title: **Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

**Abstract: **Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. In order to study their scaling and local limits, we introduce a further new family of discrete objects, called *coalescent-walk processes* and we relate them with the other previously mentioned families introducing some new bijections.

We prove joint Benjamini-Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random measure of the unit square, called the *coalescent Baxter permuton* and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations.

To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a random *continuous coalescent-walk process* encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections with the results of Gwynne, Holden, Sun (2016) on scaling limits of plane bipolar triangulations (in the Peanosphere topology). In particular, we prove that uniform bipolar orientations converge to the *uniform finite-volume bipolar map*. This proves the Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019) without restricting to triangulations.

We further present some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.

**Tuesday, April 7, 2020, 3:30-4:30**

**Videoconference Link:** Zoom

**Speaker:** Jon Peterson, Purdue

**Title: **Convergence of excited random walks on Markovian cookie stacks to Brownian motion perturbed at extrema

**Abstract: **Excited random walks are a class of self-interacting random walks where the transition probabilities depend on the local time of the walk at the present site. It has previously been shown, only for certain special cases, that when the walk is recurrent the limiting distribution of the rescaled path of the walk is a Brownian motion perturbed at its extrema. We extend this convergence to a much more general class of excited random walks: excited random walks on Markovian cookie stacks. For this we develop a completely new approach to proving convergence to perturbed Brownian motion. Previously, it was already know that generalized Ray-Knight Theorems for the excited random walk were consistent with convergence to perturbed Brownian motion. We use improved versions of these generalized Ray-Knight Theorems to construct a coupling of the random walk with a perturbed Brownian motion.

This talk is based on joint work with Elena Kosygina and Tom Mountford.

**Tuesday, April 21, 2020, 3:30-4:30**

**Videoconference Link:** Zoom

**Speaker:** Jim Gatheral, Baruch

**Title: **Diamond trees, forests, cumulants, and martingales

**Abstract: **We define a diamond operator on the space of continuous semimartingales. Compositions of diamond operations lead to trees and forests of trees. We show how forests of trees can be identified with cumulants, giving explicit expressions for cumulants. As an application, we show how to compute all terms in an expansion of the Lévy area. By reordering the trees in our cumulant expansion according to number of leaves, we retrieve our earlier exponentiation theorem, applications of which include the extension of the Bergomi-Guyon expansion to all orders in volatility of volatility. Finally, we compute exact expressions under rough Heston, in particular a closed-form expression for the leverage swap.