## Seminar Schedule:

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

**Time: September 6, 4:30 – 5:30 pm EDT**

**Speaker:**Matt Junge

**Location:**The Graduate Center, Room 9207

**Title:**Ballistic Annihilation

**Abstract:**In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

**Time: September 13**

***No seminar***

**Time: September 20, 4:30 – 5:30 pm EDT**

**Speaker:**Elena Kosygina

**Location:**The Graduate Center, Room 7314

**Title:**From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on integers.

**Abstract:**For several models of self-interacting random walks (SIRWs) generalized Ray-Knight theorems for edge local times are a very useful tool for studying the limiting distributions of these walks. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss recent results (joint work with Thomas Mountford, EPFL, and Jon Peterson, Purdue University) which, in particular, resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the uniquecandidate in the class of all perturbed Brownian motions. This negative result was somewhat unexpected. Conjectures on whether there is a suitable limiting process in this case and, if yes, what is might be are welcome.

**Time: September 27, 4:30 – 5:30 pm EDT**

***No classes: No seminar***

**Time: October 4, 4:30 – 5:30 pm EDT**

***No classes: No seminar***

**Time: October 11, 4:30 – 5:30 pm EDT**

**Speaker:**Boris Hanin

**Location:**The Graduate Center, Room 4102

**Title:**Universality in Random Neural Networks

**Abstract:**A neural network f(x;θ) is a family of functions in a variable x varying continuously in a parameter vector θ. An important chapter in the theory of neural networks is the analysis of random neural networks given by choosing θ at random. In this talk, I will focus on the simplest setting of fully connected networks for which the structure of the function f(x;θ) is described by two integer parameters, a width n and a depth L, as well as a non-linear function σ. When σ is the identity, a random fully connected network is a product of L iid random matrices of size n x n. For general σ, the correlation functions of f(x;θ) are non-linear generalization of linear statistics for such matrix products. After giving some intuition for random neural networks, I will state a range of new results regarding a novel kind of universality for their correlation functions. This notion of universality differs from the typical use of this term in random matrix theory. I will also state several conjectures about scaling limits of random neural networks as n,L tend to infinity at a fixed ratio. These conjectures are open even for random matrix products.

**Time: October 18, 5:00 – 6:00 pm EDT**(Note: different time)

**Speaker:**Yanghui Liu

**Location:**The Graduate Center, Room 4102

**Title:**Optimal rate and limit theorem for rough volatility

**Abstract:**In recent years, there has been substantive empirical evidence that volatility is “rough”. In other words, the local behavior of assets volatility is much more irregular than semi-martingales and resembles that of a fractional Brownian motion with Hurst parameter H < 0.5. This raises an intriguing question: How rough is volatility? Can we estimate the degree of irregularity H, and if so with which accuracy? In this talk, we provide a complete answer to this question. Precisely, we derive a consistent and asymptotically mixed normal estimator of H based on high-frequency price observations. In contrast to previous works, we work in a nonparametric setting and do not assume a priori relationship between volatility estimators and true volatility. Our estimator attains a rate of convergence 1/(4H+2), which is known to be optimal in a minimax sense in parametric rough volatility models. The talk does not require knowledge of mathematical finance.

This talk is based on a joint work with Chong, Hoffmann, Rosenbaum and Szymanski.

**Time: October 25, 4:30 – 5:30 pm EDT**

**Speaker:**Ben McKenna

**Location:**The Graduate Center, Room 4102

**Title:**Extremal statistics of quadratic forms of GOE/GUE eigenvectors

**Abstract:**We consider quadratic forms evaluated at GOE/GUE eigenvectors, like those studied in the context of quantum unique ergodicity. Under a rank assumption, we show that, in order to compute their extremal statistics, it suffices to replace the eigenvectors with independent Gaussian vectors. By carrying out some representative Gaussian computations, we thus find Gumbel and Weibull limiting distributions for the original problem. Joint work with László Erdős.

**Time: November 1, 4:30 – 5:30 pm EDT**

**Speaker:**Leonid Koralov

**Location:**The Graduate Center, Room 7314

**Title:**Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems and Metastability.

**Abstract:**We study diffusion processes in a bounded domain with absorbing or reflecting boundary. The generator of the process is assumed to contain two terms: the main

term that degenerates on the boundary in a direction orthogonal to the boundary and a small non-degenerate perturbation. Understanding the behavior of such processes allows us to study the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics

of solutions, as the size of the perturbation tends to zero, depends on the time scale. Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions will be considered.

**Time: November 8, 4:30 – 5:30 pm EDT**

**Speaker:**Jiaqi Liu

**Location:**The Graduate Center, room 4102

**Title:**Yaglom-type limits for branching Brownian motion with absorption in the slightly subcritical regime

**Abstract:**Branching Brownian motion is a random particle system that incorporates both the tree-like structure and the diffusion process. In this talk, we consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motion with drift, undergo dyadic fission at a constant rate, and are killed upon hitting the origin. We are interested in the asymptotic behaviors of the process conditioned on survival up to a large time t as the process approaches criticality. Results like this are called Yaglom type results. Specifically, we will discuss the construction of the Yaglom limit law, Yaglom-type limits for the number of particles and the maximal displacement. Based on joint work with Julien Berestycki, Bastien Mallein and Jason Schweinsberg.

**Time: November 15, 4:30 – 5:30 pm EDT**

***NEPS week: No seminar***

**Time: November 22, 4:30 – 5:30 pm EDT**

**Speaker:**Hanbaek Lyu

**Location:**The Graduate Center, room 4102

**Title:**Scaling limit of soliton statistics of a multicolor box-ball system

**Abstract:**The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connections between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of the classical Takahashi-Satsuma BBS as well as the multicolor BBS with one-sided random initial configurations. The integrability of BBS in this setting allows one to read-off the final soliton statistics directly from the initial configuration through various combinatorial operations. For the Takahashi-Satsuma case, these include hill-flattening operations of carrier process for soliton numbers and Pitman’s water level process for soliton lengths. For the multicolor case, we use a modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz to extract the corresponding soliton statistics.

**Time: November 29, 4:30 – 5:30 pm EDT**

**Speaker:**Augusto Teixeira

***Online seminar***

**Title:**Phase transition for percolation with axes-aligned defects

**Abstract: **In this talk we will review a model that was first introduced by Jonasson, Mossel and Peres. Starting with the usual square lattice on Z^2, entire rows (respectively columns) of edges extending along the horizontal (respectively vertical) direction are removed independently at random. On the remaining thinned lattice, Bernoulli bond percolation is performed, giving rise to a percolation model with infinite range dependencies under the annealed law. In 2005, Hoffman solved the main conjecture around this model: proving that this percolation process indeed undergoes a nontrivial phase transition. In this talk, besides reviewing this surprisingly challenging problem, we will present a novel proof, which replaces the dynamic renormalization presented previously by a static version. This makes the proof easier to follow and to extend to other models. We finally present some remarks on the sharpness of Hoffman’s result as well as a list of interesting open problems that we believe can provide a renewed interest in this family of questions.

This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

**Time: December 6, 4:30 – 5:30 pm EDT**

**Speaker:**Ron Peled

**Location:**The Graduate Center, room 4102

**Title:**Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

**Abstract: **First-passage percolation studies the geometry obtained from a random perturbation of Euclidean geometry. In the discrete planar setting, one assigns random, independent and identically distributed, lengths to the edges of the lattice Z^2 and studies the resulting geodesics – paths of minimal length between points. While the physics literature presents an elaborate picture for the behavior of the model, placing it in the KPZ universality class, mathematical progress remains rather limited. Aside from the random geometry perspective, the model also enjoys close ties with disordered spin systems.

The talk will give an introduction to planar first-passage percolation, followed by a description of recent progress, joint with Barbara Dembin and Dor Elboim, on the problem of coalescence of geodesics. Our result shows, under mild assumptions, that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result further implies the first quantitative progress on a problem of Benjamini-Kalai-Schramm (2002), which asks to prove that geodesics typically do not pass at the midpoint of the straight line segment connecting their endpoints.

No prior knowledge of first-passage percolation will be assumed.

**Time: December 13, 4:30 – 5:30 pm EDT**

**Speaker:**David Harper

***Online seminar***

**Title: **Exceptional events in critical *2d* first-passage percolation

**Abstract:**In first-passage percolation (FPP), we let τ

_{v}be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If

*F*is the distribution function of τ

_{v}, there are different regimes: if

*F(0)*is small, this weight typically grows like a linear function of the distance, and when

*F(0)*is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results that show that if the sum of

*F*diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of

^{ -1}(1/2+1/(2^{k}))*k*converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when the weight of a long path is unusually small by considering an analogous construction to Kesten’s incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.-K. Lam.

^{7/8}F^{ -1}(1/2+1/(2^{k}))