# Probability Seminar 2019

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 22, 2019 at 15:00, Penn (DRL 4C8)
One-point function estimates and natural parametrization for loop-erased random walk in three dimensions

Xinyi Li, University of Chicago

In this talk, I will talk about loop-erased random walk (LERW) in three dimensions. I will first give an asymptotic estimate on the probability that 3D LERW passes a given point (commonly referred to as the one-point function). I will then talk about how to apply this estimate to show that 3D LERW as a curve converges to its scaling limit in natural parametrization. If time permits, I will also talk about the asymptotics of non-intersection probabilities of 3D LERW with simple random walk. This is a joint work with Daisuke Shiraishi (Kyoto).

• Tuesday January 29, 2019 at 15:00, Temple (Wachman 617)
Fractional Gaussian fields in geometric quantization and the semi-classical analysis of coherent states

Alexander Moll, Northeastern University

The Born Rule (1926) formalized in von Neumann's spectral theorem (1932) gives a precise definition of the random outcomes of quantum measurements as random variables from the spectral theory of non-random matrices. In [M. 2017], the Born rule provided a way to derive limit shapes and global fractional Gaussian field fluctuations for a large class of point processes from the first principles of geometric quantization and semi-classical analysis of coherent states. Rather than take a point process as a starting point, these point process are realized as auxiliary objects in an analysis that starts instead from a classical Hamiltonian system with possibly infinitely-many degrees of freedom that is not necessarily Liouville integrable. In this talk, we present these results with a focus on the case of one degree of freedom, where the core ideas in the arguments are faithfully represented.

• Tuesday February 5, 2019 at 15:00, Penn (DRL 4C8)
Conformal embedding and percolation on the uniform triangulation

Xin Sun, Columbia University

Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice has been achieved. Smirnov’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk, I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. I will also explain how this result fits in the larger picture of random planar maps and Liouville quantum gravity.

• Tuesday February 19, 2019 at 15:00, Penn (DRL 4C8)
Asymptotic zero distribution of random polynomials

Duncan Dauvergne, University of Toronto

It is well known that the roots of a random polynomial with i.i.d. coefficients tend to concentrate near the unit circle. In particular, the zero measures of such random polynomials converge almost surely to normalized Lebesgue measure on the unit circle if and only if the underlying coefficient distribution satisfies a particular moment condition. In this talk, I will discuss how to generalize this result to random sums of orthogonal (or asymptotically minimal) polynomials.

• Tuesday February 26, 2019 at 15:00, Penn (DRL 4C8)
Distances between random orthogonal matrices and independent normals

Tiefeng Jiang, University of Minnesota

We study the distance between Haar-orthogonal matrices and independent normal random variables. The distance is measured by the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. Optimal rates are obtained. This is a joint work with Yutao Ma.

• Tuesday March 19, 2019 at 15:00, Penn (DRL 4C8)
Delocalization of random band matrices

Fan Yang, UCLA

We consider Hermitian random band matrices $H$ in dimension $d$, where the entries $h_{xy}$, indexed by $x,y \in [1,N]^d$, vanish if $|x-y|$ exceeds the band width $W$. It is conjectured that a sharp transition of the eigenvalue and eigenvector statistics occurs at a critical band width $W_c$, with $W_c=\sqrt{N}$ in $d=1$, $W_c=\sqrt{\log N}$ in $d=2$, and $W_c=O(1)$ in $d\ge 3$. Recently, Bourgade, Yau and Yin proved the eigenvector delocalization for 1D random band matrices with generally distributed entries and band width $W\gg N^{3/4}$. In this talk, we will show that for $d\ge 2$, the delocalization of eigenvectors in a certain averaged sense holds under the condition $W\gg N^{2/(2+d)}$. Based on joint work with Bourgade, Yau and Yin.

• Tuesday March 26, 2019 at 15:00, Penn (DRL 4C8)
Large deviations for functionals of Gaussian processes

Xiaoming Song, Drexel University

We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-\beta}$ with $\beta\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.

• Tuesday April 2, 2019 at 15:00, Temple (Wachman 617)
Geometry of the corner growth model

The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).

• Tuesday April 9, 2019 at 15:00, Temple (Wachman 617)
Eigenvectors of non-Hermitian random matrices

Guillaume Dubach, Courant Institute, NYU

Eigenvectors of non-Hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables also quantify the stability of the spectrum, and characterize the joint eigenvalue increments under Dyson-type dynamics. Overlaps first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices.

• Tuesday April 16, 2019 at 15:00, Temple (Wachman 617)
Stochastic homogenization for reaction-diffusion equations

Jessica Lin, McGill University

I will present several results concerning the stochastic homogenization for reaction-diffusion equations. We consider reaction-diffusion equations with nonlinear, heterogeneous, stationary-ergodic reaction terms. Under certain hypotheses on the environment, we show that the typical large-time, large-scale behavior of solutions is governed by a deterministic front propagation. Our arguments rely on analyzing a suitable analogue of “first passage times” for solutions of reaction-diffusion equations. In particular, under these hypotheses, solutions of heterogeneous reaction-diffusion equations with front-like initial data become asymptotically front-like with a deterministic speed. This talk is based on joint work with Andrej Zlatos.

• Tuesday April 30, 2019 at 15:00, Temple (Wachman 617)
The geometry of the last passage percolation problem

Tom Alberts, University of Utah

Last passage percolation is a well-studied model in probability theory that is simple to state but notoriously difficult to analyze. In recent years it has been shown to be related to many seemingly unrelated things: longest increasing subsequences in random permutations, eigenvalues of random matrices, and long-time asymptotics of solutions to stochastic partial differential equations. Much of the previous analysis of the last passage model has been made possible through connections with representation theory of the symmetric group that comes about for certain exact choices of the random input into the last passage model. This has the disadvantage that if the random inputs are modified even slightly then the analysis falls apart. In an attempt to generalize beyond exact analysis, recently my collaborator Eric Cator (Radboud University, Nijmegen) and I have started using tools of tropical geometry to analyze the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model and our work to use it to produce probabilistic information.

• Tuesday September 3, 2019 at 15:00, Penn (DRL 4C8)
Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

Ewain Gwynne, University of Cambridge

We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the Gaussian free field $h$ on a planar domain $U$, there is a unique random metric $D_h =$ "$e^{\gamma h} (dx^2 + dy^2)$" on $U$ which is uniquely characterized by a list of natural axioms.

The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dubédat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield.

Based on four joint papers with Jason Miller, one joint paper with Julien Dubédat, Hugo Falconet, Josh Pfeffer, and Xin Sun, and one joint paper with Josh Pfeffer.

• Tuesday September 10, 2019 at 15:00, Temple (Wachman 617)
Semigroups for one-dimensional Schrödinger operators with multiplicative white noise

Pierre Yves Gaudreau Lamarre, Princeton University

In this talk, we are interested in the semigroup theory of continuous one-dimensional random Schrödinger operators with white noise. We will begin with a brief reminder of the rigorous definition of these operators as well as some of the problems in which they naturally arise. Then, we will discuss the proof of a Feynman-Kac formula describing their semigroups. In closing, we will showcase an application of this new semigroup theory to the study of rigidity (in the sense of Ghosh-Peres) of random Schrödinger eigenvalue point processes.

Some of the results discussed in this talk are joint work with Promit Ghosal (Columbia) and Yuchen Liao (Michigan).

• Tuesday September 24, 2019 at 15:00, Temple (Wachman 617)
TBA

Axel Saenz, University of Virginia

• Tuesday October 1, 2019 at 15:00, Penn (DRL 4C8)
TBA

Amir Dembo, Stanford University

• Tuesday October 8, 2019 at 15:00, Temple (Wachman 617)
TBA

Li-Cheng Tsai, Rutgers University

• Tuesday October 15, 2019 at 15:00, Penn (DRL 4C8)
TBA

Tatyana Shcherbina, Princeton University

• Tuesday October 29, 2019 at 15:00, Penn (DRL 4C8)
TBA

Eliran Subag, Courant Institute, NYU

• Tuesday November 5, 2019 at 15:00, Temple (Wachman 617)
TBA

Michael Damron, Georgia Tech

• Tuesday November 19, 2019 at 15:00, Temple (Wachman 617)
TBA

Yu Gu, Carnegie Mellon University

• Tuesday December 3, 2019 at 15:00, Penn (DRL 4C8)
TBA

Eyal Lubetzky, Courant Institute, NYU