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.

*Shirshendu Chatterjee (http://shirshendu.ccny.cuny.edu/*)

## Seminar Schedule:

**Tuesday, , 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Upcoming Talks**

**Time: December 08, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: Julien Berestycki, Oxford**

**Title: The distance between the two branching Brownian motion leaders **

**Abstract: **Let $d_{12}$ be the distance between the two rightmost particles in a branching Brownian motion at large times. Brunet and Derrida have shown that the distribution of this variable can be obtained through the long-time behaviour of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of the tail of $d_{12}$ which were previously known only up to “exponential order;” we discover an algebraic correction to this behavior. Based on a joint work with Eric Brunet, Graham Cole, Leonid Mytnik, Jeam-Michel Roquejoffre and Lenya Ryzhik

**Previous Talks**

**Time: September 01, 3:30 – 4:30 pm EST**

**Speaker: Matthew Junge; Baruch College; Department of Mathematics, CUNY**

**Title: Modeling COVID-19 Spread in Small Colleges**

**Abstract: Many colleges are reopening amid Fall 2020 of the COVID-19 pandemic with extreme measures in place: testing, dedensification, building closures, among others. We develop an agent-based network model to test intervention effectiveness. Our focus is on small colleges, which in aggregate serve over one million U.S. students, and have not been considered in-depth by existing models. We will survey how COVID-19 predictions are made for large areas like countries and cities, then go into detail about the models that came out this summer for disease spread on college campuses. From there, we will describe our model and findings. One of the more striking findings suggests that building closures may have unintended negative consequences. This is part of a broader observation that how students conduct themselves will determine if they get to enjoy, albeit a bit differently, the benefits of college life, or pass another year learning from a screen in their bedroom. Preprint available at https://arxiv.org/abs/2008.09597.**

**Time: September 08, 4:30 – 5:30 pm EST**

**Speaker: Emma Bailey; Graduate Center, CUNY**

**Title: Branching random walks and moments of moments**

**Abstract: Recently there has been a great deal of interest in understanding the moments of partition functions of logarithmically correlated processes. In this talk, I will present results for the moments of the partition function for a branching random walk on a binary tree of depth n with Gaussian weightings. We obtain explicit formulae for the first few moments, and in the large n limit, our expression coincides with recent conjectures and results for the moments of moments of characteristic polynomials of random unitary matrices. This is joint work with Jon Keating.
**

**Time: September 15, 4:45 – 5:45 pm EST**

**Speaker: Yanghui Liu; Baruch College, CUNY**

**Title: **Discrete rough paths and numerical approximations.

**Abstract: In this talk, I will focus on a series of results concerning the numerical approximation of rough integration and rough differential equations, as well as the Malliavin differentiability and weak convergence problems of numerical schemes. I will explain the links between numerical approximations and weighted random sums, and show how to transfer limits taken on a Gaussian signature to limits involving controlled processes, by means of the typical expansions of the rough-paths theory. I will try to introduce most of my notations carefully, and keep the needed stochastic analysis to a minimum level. The talk does not require the rough-path background.**

**Time: September 22, 4:30-5:30 pm EST**

**Speaker: Wai Tong (Louis) Fan, Indiana University, Bloomington.**

**Title: **Stochastic PDE as scaling limits of interacting particle systems

**Abstract: **Interacting particle models are often employed to gain an understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as the efficiency and robustness of macroscopic models.

In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics.

**Time: October 06, 3:30 – 4:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: Winston Heap; Max Planck Institute**

**Title: Random multiplicative functions and a model for the Riemann zeta function**

**Abstract**: We look at a weighted sum of random multiplicative functions and view this as a model for the Riemann zeta function. We investigate various aspects including its high moments, distribution and maxima.

**Time: October 13, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: Sayan Banerjee, University of North Carolina, Chapel Hill.**

**Title: Persistence and root detection algorithms in growing networks **

**Abstract: Motivated by questions in Network Archaeology, we investigate statistics of dynamic networks that are persistent, that is, they fixate almost surely after some random time as the network grows. We consider generalized attachment models of network growth where at each time n, an incoming vertex attaches itself to the network through mn edges attached one-by-one to existing vertices with probability proportional to an arbitrary function f of their degree. We identify the class of attachment functions f for which the maximal degree vertex persists and obtain asymptotics for its index when it does not. We also show that for tree networks, the centroid of the tree persists and use it to device root-finding algorithms and quantify their efficacy. **

**This is joint work with Shankar Bhamidi. **

**Time: October 20, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: **Patricia Alonso Ruiz, Texas A&M University

**Title: Short-time analysis of heat kernels on diamond fractals
**

**Abstract: **The heat kernel of a diffusion process can be understood as the associated transition probability density function. Its short-time behavior encodes geometric properties of the underlying space and is of particular interest when the latter is highly non-smooth.

In this talk, we discuss diffusion processes on a parametric family of fractals called generalized diamond fractals. They arise as scaling limits of diamond hierarchical lattices studied in the physics literature in relation to random polymers, Ising and Potts models.

These spaces have some unusual features: the volume doubling property is not satisfied, and for many parameter choices the corresponding diamond fractal is not self-similar. However, its structure as projective limit will allow us to derive estimates for the heat kernel and the regularity of the associated heat semigroup.

**Time: October 27, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: Markus Heydenreich,Mathematisches Institut
der Universität München**

**Title: **The weight-dependent random connection model

**Abstract: **We investigate a large class of random graphs on the points of a Poisson process in Eudlidean space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. This generalises many spatial random graph models.

Our focus is on the question whether infinite components are recurrent or transient, and we demonstrate that the answer depends on the model parameters.

In a plain version of the random connection model, where weights are ignored, we can even analyse the model at the phase transition point. Indeed, we obtain an infrared bound for the critical connectivity function if the dimension is sufficiently large or if the pair connection function has sufficiently slow decay. This is achieved through an adaptation of the percolation lace expansion for Poisson processes.

Based on joint work with Peter Gracar, Remco van der Hofstad, Günter Last, Kilian Matzke, Christian Mönch, and Peter Mörters.

**Time: November 03, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: **Swee Hong Chan, UCLA

**Title: **Sorting probability for Young diagrams

**Abstract: **Can you always find two elements x, y of a partially ordered set, such that, the probability that x is ordered before y when the poset is ordered randomly, is between 1/3 and 2/3? This is the celebrated 1/3 – 2/3 Conjecture, which has been called “one of the most intriguing problems in the combinatorial theory of posets”.

We will explore this conjecture for posets that arise from (skew-shaped) Young diagrams, where total orderings of these posets correspond to standard Young tableaux. We will show that that these probabilities are arbitrarily close to 1/21/2, by using random walk estimates and the state-of-the-art hook-length formulas of Naruse.

This is a joint work with Igor Pak and Greta Panova.

**Time: November 10, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: Gourab Ray, University of Victoria**

**Title: Characterizing Gaussian free field with low moments**

**Abstract:** Gaussian free field is a Gaussian field which belongs in a family of fractional Gaussian fields. This talk will be about the continuum Gaussian free field in 2 dimensions, which is a canonical object as it is believed to be the scaling limit of the fluctuations of many natural statistical physics models (in this sense, it is a cousin of Brownian motion in 1 dimension). There are many characterization theorems for Brownian motion in the literature (Levy characterization being perhaps the most classical one). However for Gaussian free field, this area is quite unexplored.

In this talk, I will describe a characterization theorem for the Gaussian free field, namely any conformally invariant field which satisfies a domain Markov property must be the Gaussian free field. In a previous work with Nathanael Berestycki and Ellen Powell, we proved this characterization theorem under the additional assumption of finite fourth moments. In a more recent work (with the same set of authors), we reduce the moment condition to 1+\eps. I will give an overview of the previous work, and describe how reducing the moments require a non-trivial averaging procedure which could be of independent interest.

**Time: December 01, 4:30-5:30 pm EST**

**Videoconference Link: Zoom**

**Speaker: Xiaoqin Guo, University of Cincinnati**

**Title: Optimal rates of periodic homogenization of linear elliptic non-divergence form equations**

**Abstract: **In this talk, we consider the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We will also discuss its relation to large scale behavior of random walks in a periodic environment. Our conclusion is that generically, the optimal rate is of order O(epsilon). This is based on joint work with Hung V. Tran (Madison) and Yifeng Yu (Irvine).