# Probability Seminar

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday January 25, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Strong Quantum Unique Ergodicity and its Gaussian fluctuations for Wigner matrices

Giorgio Cipolloni, Princeton University

We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis, which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound as an input, we analyze the Stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in the QUE.

The main methods behind the above results are:

(i) multi-resolvent local laws established via a novel bootstrap scheme;

(ii) energy estimates for SEE.

• Tuesday February 1, 2022 at 15:30, Temple (Wachman Hall 617)
On convergence of the cavity and Bolthausen’s TAP iterations to the local magnetization

Si Tang, Lehigh University

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work, Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen’s iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen’s scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated. This is a joint work with Wei-Kuo Chen (University of Minnesota).

• Tuesday February 8, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Lozenge tilings and the Gaussian free field on a cylinder

Marianna Russkikh, MIT

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.

• Tuesday February 15, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Scaling limits of the Laguerre unitary ensemble

Xuan Wu, University of Chicago

In this talk, we will discuss the LUE, focusing on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This novel Gibbsian line ensemble enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler.

• Tuesday February 22, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
The local environment of a geodesic in Last-Passage Percolation

Lingfu Zhang, Princeton University

In exponential Last-Passage Percolation, each vertex in the 2D lattice is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. This model was first introduced to model fluid flow through a random medium. It is also a central model in the KPZ universality class and related to various natural processes.

A classical question asks what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present new results on the convergence of the ‘environment’ as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure. In addition, we obtain an explicit description of the limiting ‘environment’. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples.

This is based on joint work with James Martin and Allan Sly.

• Tuesday March 8, 2022 at 15:30, Temple (Wachman Hall 617)
Large deviation estimates for Selberg’s central limit theorem and applications

Emma Bailey, The Graduate Center, CUNY

Selberg’s celebrated central limit theorem shows that the logarithm of the zeta function at a typical point on the critical line behaves like a complex, centered Gaussian random variable with variance $\log\log T$. This talk will present recent results showing that the Gaussian decay persists in the large deviation regime, at a level on the order of the variance, improving on the best known bounds in that range.  We also present various applications, including on the maximum of the zeta function in short intervals. Whilst the results are number theoretic, the tools used are predominantly probabilistic in nature.  This work is joint with Louis-Pierre Arguin.

• Tuesday March 15, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Metric growth dynamics in Liouville quantum gravity

Hugo Falconet, Courant Institute, NYU

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.

• Tuesday March 22, 2022 at 15:30, Temple (Wachman Hall 617)
From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on integers

Elena Kosygina, Baruch College & The Graduate Center, CUNY

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 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 unique candidate in the class of 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 it might be are welcome.

• Tuesday March 29, 2022 at 15:30, Temple (Wachman Hall 617)
Localization and delocalization in Erdős–Rényi graphs

Johannes Alt, Courant Institute, NYU

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.

• Tuesday April 5, 2022 at 15:30, Temple (Wachman Hall 617)
Uniqueness in Cauchy problems for diffusive real-valued strict local martingales

Kasper Larsen, Rutgers University

For a real-valued one-dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution. We exemplify our results using quadratic normal volatility models and the two-dimensional Bessel process. Joint work with Umut Cetin (LSE).

• Tuesday April 12, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Six-vertex model and the KPZ universality class

Amol Aggarwal, Columbia University

In this talk, we explain recent results relating the six-vertex model and the Kardar-Parisi-Zhang (KPZ) universality class. In particular, we describe how the six-vertex model can be used to analyze stochastic interacting particle systems, such as asymmetric exclusion processes, and how infinite-volume pure states of the ferroelectric six-vertex model exhibit fluctuations of order $N^{1/3}$, a characteristic feature of systems in the KPZ universality class.

• Tuesday April 19, 2022 at 15:30, Temple (Wachman Hall 617)
Understanding the upper tail behaviour of the KPZ equation via the tangent method

Milind Hegde, Columbia University

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.

• Tuesday April 26, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Multiplicative chaos of the Brownian loop soup

Antoine Jego, MSRI

On the one hand, the 2D Gaussian free field (GFF) is a log-correlated Gaussian field whose exponential defines a random measure: the multiplicative chaos associated to the GFF, often called Liouville measure. On the other hand, the Brownian loop soup is an infinite collection of loops distributed according to a Poisson point process of intensity $\theta$ times a loop measure. At criticality ($\theta = 1/2$), its occupation field is distributed like half of the GFF squared (Le Jan's isomorphism). The purpose of this talk is to understand the infinitesimal contribution of one loop to Liouville measure in the above coupling. This work is not restricted to the critical intensity and provides the natural notion of multiplicative chaos associated to the Brownian loop soup when $\theta$ is not equal to $1/2$.

• Tuesday September 6, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Fractal Geometry of the KPZ equation

Promit Ghosal, MIT

The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to the KPZ universality class. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention a work in progress with Jaeyun Yi for the (2+1)-d case.

• Tuesday September 13, 2022 at 15:30, Temple (Wachman Hall 617)
Ballistic annihilation

Matthew Junge, Baruch College

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.

• Tuesday September 20, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
A central limit theorem for square ice

Wei Wu, NYU Shanghai

In the area of statistical mechanics, an important open question is to show that the height function associated with the square ice model (i.e., planar six vertex model with uniform weights), or equivalently with uniform graph homeomorphisms, converges to a continuum Gaussian free field in the scaling limit. I will review some recent results about this model, including that the single point height function, upon renormalization, converges to a Gaussian random variable.

• Tuesday September 27, 2022 at 15:30, Temple (Wachman Hall 617)
On roots of random trigonometric polynomials and related models

Hoi Nguyen, Ohio State University

In this talk, we will discuss various basic statistics of the number of real roots of random trigonometric polynomials, as well as the minimum modulus and the nearest roots statistics to the unit circle of Kac polynomials. We will emphasize the universality aspects of all these problems.

Based on joint works with Cook, Do, O. Nguyen, Yakir and Zeitouni.

• Tuesday October 4, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Yaglom-type limits for branching Brownian motion with absorption in the slightly subcritical regime

Jiaqi Liu, Penn

Branching Brownian motion is a random particle system which 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 talk about 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.

• Tuesday October 11, 2022 at 15:30, Temple (Wachman Hall 617)
Ergodicity and synchronization of the KPZ equation

Chris Janjigian, Purdue University

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some results-in-progress which show that these distributions are totally ergodic and present some progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. The talk will discuss our coupling of Hopf-Cole solutions, which enables us to study the KPZ equation started from any measurable function valued initial condition. Through this coupling, we give a sharp characterization of when such solutions explode, show that all non-explosive functions become instantaneously continuous, and then study the problem of ergodicity on a natural topology on the space of non-explosive continuous functions (mod constants) in which the equation defines a Feller process. We show that any ergodic stationary distribution on this space is either a Brownian motion with drift or a process of a very peculiar form which will be described in the talk.

Based on joint works with Tom Alberts, Firas Rassoul-Agha, and Timo Seppäläinen.

• Tuesday October 18, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Graph limits and graph homomorphism density inequalities

Fan Wei, Princeton University

Graph limits is a recently developed powerful theory in studying large (weighted) graphs from a continuous and analytical perspective. It is particularly useful when studying subgraph homomorphism density, which is closely related to graph property testing, graph parameter estimation, and many central questions in extremal combinatorics. In this talk, we will show how the perspective of graph limits helps with graph homomorphism inequalities and how to make advances in a common theme in extremal combinatorics: when is the random construction close to optimal? We will also show some hardness results for proving general theorems in graph homomorphism density inequalities.

• Tuesday October 25, 2022 at 15:30, Temple (Wachman Hall 617)
Optimal delocalization for generalized Wigner matrices

Lucas Benigni, Université de Montréal

We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

• Tuesday November 1, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
Sandpiles

Ahmed Bou-Rabee, Cornell University

I will introduce the Abelian sandpile model and discuss its large-scale behavior in random environments and on different lattices. There are many open questions.

• Tuesday November 8, 2022 at 15:30, Temple (Wachman Hall 617)
Convergence of densities of spatial averages of the stochastic heat equation

Şefika Kuzgun, University of Rochester

Let $u$ be the solution to the one-dimensional stochastic heat equation driven by a space-time white noise with constant initial condition. The purpose of this talk is to present a recent result on the uniform convergence of the density of the normalized spatial averages of the solution $u$ on an interval $[-R,R]$, as $R$ tends to infinity, to the density of the standard normal distribution, assuming some non-degeneracy and regularity conditions on the diffusion coefficient. These results are based on the combination of Stein's method for normal approximations and Malliavin calculus techniques. This is a joint work with David Nualart.

• Tuesday November 29, 2022 at 15:30, Penn (David Rittenhouse Lab 4C8)
(at UPenn) Non-backtracking spectra of random hypergraphs and community detection

Yizhe Zhu, UC Irvine

The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs has become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al (2015). We characterize the spectrum of the non-backtracking operator for sparse random hypergraphs and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix of the hypergraph. Based on joint work with Ludovic Stephan (EPFL).