## Current Series

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Wed Sep 16

## Probability Seminar

4:00pm - Via Zoom
Tail bounds for the averaged empirical distribution on a geodesic in first-passage percolation
Wai-Kit Lam, UMN

Consider $\mathbb{Z}^d$ with nearest-neighbor edges. In first-passage percolation, we place i.i.d. nonnegative weights $(t_e)$ on the edges, and study the induced graph metric $T(x,y)$. A geodesic is a minimizing path for this metric. In a joint work with M. Damron, C. Janjigian and X. Shen, we study the empirical distribution on a geodesic $\gamma$ from $0$ to $x$: $\nu^x(B) := (number of edges e in \gamma with t_e \in B) / (number of edges e in \gamma)$. We establish bounds for the averaged empirical distribution $E \nu^x(B)$, particularly showing that if the law of $t_e$ has finite moments of any order strictly larger than 1, then roughly speaking the limiting averaged empirical distribution has all moments.

Wed Sep 23

## Probability Seminar

9:00am - Via Zoom
Forest fire processes and near-critical percolation with heavy-tailed impurities
Pierre Nolin, City University of Hong Kong

We discuss models of forest fires (or epidemics): on a given planar lattice, all vertices are initially vacant, and then become occupied at rate 1. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate, all the vertices connected to it "burn" instantaneously, i.e. they become vacant. We want to analyze the behavior of such processes near and beyond the critical time (the time after which, in the absence of fires, infinite connected components would emerge).

We are led to introduce a percolation model where regions ("impurities") are removed from the lattice, in an independent fashion. These impurities are not only microscopic, but also allowed to be mesoscopic. We are interested in whether the connectivity properties of percolation remain of the same order as without impurities, for values of the percolation parameter close to the critical value. This generalizes a celebrated result by Kesten for near-critical percolation (that can be viewed as critical percolation with single-site impurities).

This talk is based on a joint work with Rob van den Berg (CWI and VU, Amsterdam).

Wed Sep 30

## Probability Seminar

4:00pm - Via Zoom
Empirical measures, geodesic lengths, and a variational formula in first-passage percolation

We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them.  Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to ne_1.  For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity.  The key tool is a new variational formula for the time constant.  In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

Wed Oct 07

## Probability Seminar

4:00pm - Via Zoom
The r-to-p norm of non-negative random matrices
Souvik Dhara, MIT

For an n\times n matrix A_n, the r\to p operator norm is defined as \|A_n\|_{r\to p}:= \sup_{x \in R^n:\|x\|_r\leq 1 } \|A_n\|_p for r, p\geq 1. For different choices of r and p, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and finding oblivious routing schemes in transportation networks. This talk considers r\to p norms of symmetric random matrices with nonnegative entries, including adjacency matrices of  Erdos-Renyi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices. For 1< p\leq r< \infty, the asymptotic normality,  as n\to\infty, of the appropriately centered and scaled norm \|A_n\|_{r\to p} is established. Furthermore,  a sharp \ell_\infty-approximation for the unique maximizing vector in the definition of \|A_n\|_{r\to p} is obtained, which may be of independent interest. In fact, the vector approximation result is shown to hold for a broad class of deterministic sequence of matrices having certain asymptotic expansion properties. The results obtained can be viewed as a  generalization of the seminal results of F\"{u}redi and  Koml\'{o}s (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices, which corresponds to the special case r=p=2 considered here. In the general case with 1< p\leq r < \infty, the spectral methods are no longer applicable, which requires a new approach, involving a refined convergence analysis of a nonlinear power method and establishing a perturbation bound on the maximizing vector.

This is based on a joint work with Debankur Mukherjee (Georgia Tech) and Kavita Ramanan (Brown University).

Wed Oct 21

## Probability Seminar

4:00pm - Via Zoom
On microscopic derivation of a continuum mean-curvature flow
Sunder Sethuraman, University of Arizona

We derive a continuum mean-curvature flow as a scaling limit of a class of zero-range + Glauber interacting particle systems. The zero-range part moves particles while preserving particle numbers, and the Glauber part allows birth and death of particles, while favoring two levels of particle density. When the two parts are simultaneously seen in certain (different) time-scales, and the Glauber part is bi-stable', a mean-curvature interface flow, incorporating a homogenized surface tension' reflecting microscopic rates, between the two levels of particle density, can be captured as a limit of the mass empirical density. This is work with Perla El Kettani, Tadahisa Funaki, Danielle Hilhorst, and Hyunjoon Park.

Wed Oct 28

## Probability Seminar

4:00pm - Via Zoom
Random walks on dynamic random environments with non-uniform mixing
Marcelo Hilário, The Federal University of Minas Gerais

In this talk, we will discuss recent results on the limiting behavior of random walks on dynamic random environments. We will mainly discuss the case when then random walk evolves on one-dimensional random environments given by conservative interacting particle systems such as the simple symmetric exclusion process. Our results depend a great deal on space-time mixing properties imposed on the underlying environment and also on other features like the dimension and the type of allowed transitions. Conservation of particles leads to poor-mixing conditions which complicate the applicability of available tools and to overcome this difficulty we use renormalization to obtain the law of large numbers, large deviation estimates, and sometimes central limit theorems. The talk is based on several joint works with Oriane Blondel, Frank den Hollander, Daniel Kious, Renato dos Santos, and Vladas Sidoravicius.

Wed Nov 04

## Probability Seminar

4:00pm - Via Zoom
Hypercontractivity, Convexity, and Lower Deviations
Petros Valettas, University of Missouri

The concentration of measure phenomenon is an indispensable tool in the study of high-dimensional phenomena.
Nonetheless, there exist several key situations that yields suboptimal results. We will discuss how this probabilistic principle admits stronger forms in the presence of convexity and how the local version of them can be combined with a blend of analytic, combinatorial, and topological methods in order to obtain sharp small ball probabilities for norms in high dimensions. Time permitting we will present applications in asymptotic geometric analysis. Based on joint work with Grigoris Paouris and Konstantin Tikhomirov.

Wed Nov 18

## Probability Seminar

9:00am - Via Zoom
Quantitative estimates for the effect of disorder on low-dimensional lattice systems
Ron Peled, Tel Aviv University

The addition of an arbitrarily weak random field to low-dimensional classical statistical physics models leads to the "rounding" of first-order phase transitions at all temperatures, as predicted in 1975 by Imry and Ma and proved rigorously in 1989 by Aizenman and Wehr. This phenomenon was recently quantified for the two-dimensional random-field Ising model (RFIM), proving that it exhibits exponential decay of correlations at all temperatures. The RFIM analysis relies on monotonicity (FKG) properties which are absent in many other classical models. The talk will present new results on the quantitative aspects of the phenomenon for general systems with discrete and continuous symmetries, including Potts, spin O(n), spin glass and height function models.
Joint work with Paul Dario and Matan Harel.

Wed Dec 02

## Probability Seminar

9:00am - Via Zoom
Some properties of the discrete membrane model
Alessandra Cipriani, UT Delft

The discrete membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is a very close relative of the discrete Gaussian free field (DGFF), for which instead the most likely interfaces are those preserving the mean height. However working with the two models presents some key differences, in that in the MM the shape is driven by the biharmonic operator, while the DGFF is essentially a Gaussian perturbation of harmonic functions. In particular, a lot of tools (electrical networks, random walk representation of the covariance) are available for the DGFF and lack in the MM. In this talk we will review some basic properties of the MM, and we will investigate a random walk representation for the covariances of the MM and what it can bring forth in terms of scaling limits of its extremes.

This talk is based on joint works, partly ongoing, with Biltu Dan, Rajat Subhra Hazra (ISI Kolkata) and Rounak Ray (TU/e).

Wed Dec 09

## Probability Seminar

4:00pm - Via Zoom
On the extension complexity of random polytopes
Lisa Sauermann, IAS

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random polytopes. For a fixed dimension d, we consider random d-dimensional polytopes obtained as the convex hull of independent random points either in the unit ball or on the unit sphere. In both cases, we prove that the extension complexity is typically on the order of the square root of number of vertices of the polytope. Joint work with Matthew Kwan and Yufei Zhao

Wed Jan 20

## Probability Seminar

4:00pm - via Zoom
Grothendieck L_p problem for Gaussian matrices
Arnab Sen, UMN

Consider the optimization problem where we maximize the quadratic form of a large Gaussian matrix over the unit L_p ball. The case p = 2 corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble. On the other hand, when p = ?,the maximum value is the ground state energy of the mean-field Ising spin glass model and its limit can be expressed by the Parisi formula. In the talk, I will describe the limit of this optimization problem for general p and discuss some results on the behavior of optimizers along with some open problems.

This is joint work with Wei-Kuo Chen.

Wed Jan 27

## Probability Seminar

9:00am - via Zoom
The two-dimensional continuum random field Ising model
Rongfeng Sun, National University of Singapore

In this talk, I will explain how to construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman. Based on joint work with Adam Bowditch.

Wed Feb 03

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Feb 10

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Feb 17

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Feb 24

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Mar 03

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Mar 10

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Mar 17

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Mar 24

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Mar 31

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Apr 07

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Apr 14

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD
Wed Apr 21

## Probability Seminar

4:00pm - via Zoom
Probability Seminar
TBD