# Probability Seminar 2020

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding and Robin Pemantle (Penn).

Talks are Tuesdays 3:00 - 4:00 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 21, 2020 at 15:00, Penn (DRL 4C8)
Fast randomized iterative numerical linear algebra for quantum chemistry (and other applications)

Jonathan Weare, Courant Institute, NYU

I will discuss a family of recently developed stochastic techniques for linear algebra problems involving very large matrices.  These methods can be used to, for example, solve linear systems, estimate eigenvalues/vectors, and apply a matrix exponential to a vector, even in cases where the desired solution vector is too large to store.  The first incarnations of this idea appear for dominant eigenproblems arising in statistical physics and in quantum chemistry and were inspired by the real space diffusion Monte Carlo algorithm which has been used to compute chemical ground states since the 1970's.  I will discuss our own general framework for fast randomized iterative linear algebra as well share a very partial explanation for their effectiveness.  I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.  This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, Sam Greene, and Rob Webber.

• Tuesday January 28, 2020 at 15:00, Temple (Wachman 617)
The KPZ fixed point

Konstantin Matetski, Columbia University

The KPZ universality class is a broad collection of models, which includes directed random polymers, interacting particle systems and random interface growth, characterized by unusual scale of fluctuations which also appear in the random matrix theory. The KPZ fixed point is a scaling invariant Markov process which is the conjectural universal limit of all models in the class. A complete description of the KPZ fixed point was obtained in a joint work with Jeremy Quastel and Daniel Remenik. In this talk I will describe how the KPZ fixed point was derived by solving a special model in the class called TASEP.

• Tuesday February 4, 2020 at 15:00, Penn (DRL 4C8)
Scaling limit of a directed polymer among a Poisson field of independent walks

Jian Song, Shandong University

We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb{Z}$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation with a Gaussian noise whose space-time covariance is given by the heat kernel.

• Tuesday February 11, 2020 at 15:00, Temple (Wachman 617)
Entropy of ribbon tilings

Vladislav Kargin, Binghamton University

I will talk about ribbon tilings, which have been originally introduced and studied by Pak and Sheffield. These are a generalization of the domino tilings which, unfortunately, lacks relations to determinants and spanning trees but still retains some of the nice properties of domino tilings. I will explain how ribbon tilings are connected to multidimensional heights and acyclic orientations, and present some results about enumeration of these tilings on simple regions. Joint work with Yinsong Chen.

• Tuesday February 18, 2020 at 15:00, Penn (DRL 4C8)
Extreme eigenvalue distributions of sparse random graphs

Jiaoyang Huang, Institute for Advanced Study

I will discuss the extreme eigenvalue distributions of adjacency matrices of sparse random graphs, in particular the Erdős-Rényi graphs $G(N,p)$ and the random $d$-regular graphs. For Erdős-Rényi graphs, there is a crossover in the behavior of the extreme eigenvalues. When the average degree $Np$ is much larger than $N^{1/3}$, the extreme eigenvalues have asymptotically Tracy-Widom fluctuations, the same as Gaussian orthogonal ensemble. However, when $N^{2/9}\ll Np\ll N^{1/3}$ the extreme eigenvalues have asymptotically Gaussian fluctuations. The extreme eigenvalues of random $d$-regular graphs are more rigid, we prove on the regime $N^{2/9}\ll d\ll N^{1/3}$ the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by the Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have the second-largest eigenvalue strictly less than $2\sqrt{d-1}$. These are based on joint works with Roland Bauerschmids, Antti Knowles, Benjamin Landon and Horng-Tzer Yau.

• Tuesday February 25, 2020 at 15:00, Temple (Wachman 617)
Universality of extremal eigenvalue statistics of random matrices

Benjamin Landon, MIT

The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory.  However, the behavior of certain "extremal" or "critical" observables is not fully understood.  Towards the former, we discuss progress on the universality of the largest gap between consecutive eigenvalues.  With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.

• Tuesday March 3, 2020 at 15:00, Penn (DRL 4C8)
Estimation of Wasserstein distances in the spiked transport model

Jonathan Niles-Weed, Courant Institute, NYU

We propose a new statistical model, generalizing the spiked covariance model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study various probabilistic and statistical features of this model, including the estimation of the Wasserstein distance, which we show can be accomplished by an estimator which avoids the "curse of dimensionality" typically present in high-dimensional problems involving the Wasserstein distance. However, this estimator does not seem possible to compute in polynomial time, and we give evidence that any computationally efficient estimator is bound to suffer from the curse of dimensionality. Our results therefore suggest the existence of a computational-statistical gap.

Joint work with Philippe Rigollet.

• Tuesday March 17, 2020 at 15:00, Penn (DRL 4C8)
(POSTPONED)

Robert Hough, Stony Brook University

• Tuesday March 24, 2020 at 15:00, Wachman 617
(POSTPONED)

Kavita Ramanan, Brown University

• Tuesday March 31, 2020 at 15:00, Penn (DRL 4C8)
(POSTPONED)

Yilin Wang, MIT

• Tuesday April 7, 2020 at 15:00, Temple (Wachman 617)
(POSTPONED)

Hoi Nguyen, Ohio State University

• Tuesday April 14, 2020 at 15:00, Penn (DRL 4C8)
(POSTPONED)

Louis Fan, Indiana University

• Tuesday April 21, 2020 at 15:00, Temple (Wachman 617)
(POSTPONED)

Kyle Luh, Harvard University

• Tuesday April 28, 2020 at 15:00, Penn (DRL 4C8)
(POSTPONED)

Gourab Ray, University of Victoria