# 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)
TBA

Xiaoming Song, Drexel University

• Tuesday April 2, 2019 at 15:00, Temple (Wachman 617)
TBA

• Tuesday April 9, 2019 at 15:00, Temple (Wachman 617)
TBA

Guillaume Dubach, NYU Courant

• Tuesday April 16, 2019 at 15:00, Temple (Wachman 617)
TBA

Jessica Lin, McGill University

• Tuesday April 30, 2019 at 15:00, Temple (Wachman 617)
TBA

Tom Alberts, University of Utah