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Stochastics Seminar

Click here for the Stochastics Group website

Fall 2023 Friday 3:00-4:00 PM (unless otherwise announced)

Room for in-person: JWB 208

Zoom information: E-mail the organizers

(in person talks are not broadcast on Zoom)

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Date Speaker Title (click for abstract, if available)
Friday, September 1st Firas Rassoul-Agha
University of Utah

I will present recent results on the uniqueness, ergodicity, and attractiveness of stationary solutions to the Kardar-Parisi-Zhang (KPZ) equation on the real line. It is known that this equation admits Brownian motion with a linear drift as a stationary solution. We show that these solutions are attractive, a principle known as one force--one solution (1F1S): the solution to the KPZ equation started in the distant past from an initial condition with a given velocity will converge almost surely to a Brownian motion with that same drift. As a result, we deduce that these stationary measures are in fact totally ergodic. Furthermore, we can couple all these stationary solutions so that the above attractiveness holds simultaneously (i.e. on a single full measure event) for all but a countably infinite (random) set of asymptotic velocities. This is joint work with Chris Janjigian and Timo Seppalainen. Part of the work is also joint with Tom Alberts.

Friday, September 15th Anna Little
University of Utah

Recent advances in applications such as cryo-electron microscopy have sparked increased interest in the mathematical analysis of multi-reference alignment (MRA) problems, where the goal is to recover a hidden signal from many noisy observations. The simplest model considers observations of a 1-d hidden signal which have been randomly translated and corrupted by high additive noise. This talk generalizes this classic problem by incorporating random dilations into the data model in addition to random translations and additive noise, and explores multiple approaches to its solution based on translation invariant representations. Random dilations cause large perturbations in the high frequencies, making this a challenging model. When the dilation distribution is unknown, the power spectrum of the hidden signal can be approximated by applying a nonlinear unbiasing procedure to a wavelet-based, translation invariant representation and then solving an optimization problem. When the dilation distribution is known, a more accurate unbiasing procedure can be applied directly to the empirical Fourier invariants to obtain an unbiased estimator of the Fourier invariants of the hidden signal, and the convergence rate of the estimator can be precisely quantified in terms of the sample size and noise levels. Theoretical results are supported by extensive numerical experiments on a wide range of signals. Time permitting, we will also see how these signal processing tools can be applied in the novel context of distribution learning from biased, sparse batches.

Friday, September 22nd TBA


Friday, September 29th TBA


Friday, October 6th Lee Cheuk Yin
Chinese University of Hong Kong

Local times of stochastic processes are closely related to sample path irregularities, and fractal and geometric properties of level sets. I will discuss two basic approaches for studying local times: one via Fourier transform and the other via chaos expansion. In particular, I will focus on the first approach in the case of Gaussian random fields and present some local time and related results which can be applied to the solutions of systems of linear SPDEs such as stochastic heat and wave equations. If time permits, I will briefly explain our research direction on local times of non-Gaussian processes with the second approach.

Friday, October 20th Xiao Shen
University of Utah

Many two-dimensional random growth models, including first-passage and last-passage percolation, are conjectured to fall within the KPZ universality class under mild assumptions on the underlying noise. In recent years, researchers have focused on a subset of exactly solvable models, where these conjectures can be rigorously verified. A wide array of methods has been employed, encompassing integrable probability, Gibbsian line ensemble, percolation arguments, and coupling techniques. This talk discusses a specific line of research that combines percolation arguments and coupling techniques to gain insights into the random geometry and space-time profiles of such growth models in the positive-temperature setting.

Friday, October 27th Manan Bhatia

In random geometry, a recurring theme is that all geodesics emanating from a typical point merge into each other close to their starting point, and we call such points as 1-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this talk, we will consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virág and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. The main result that we will discuss is that the above dimension is almost surely equal to 1/3. This is in contrast to Ganguly-Zhang where it was shown that the set of atypical stars on the vertical line {x=0} has dimension 2/3. This reduction of the dimension from 2/3 to 1/3 yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.

Thursday, November 2nd Han Le
University of Michigan

The spherical Sherrington--Kirkpatrick (SSK) model and its bipartite analog both exhibit the phenomenon that their free energy fluctuations are asymptotically Gaussian at high temperature but asymptotically Tracy--Widom at low temperature. This was proved in two papers by Baik and Lee, for all non-critical temperatures. The case of critical temperature was recently computed for the SSK model in two separate papers, one by Landon and the other by Johnstone, Klochkov, Onatski and Pavlyshyn. In this talk, we will discuss the critical temperature result for the bipartite SSK model. In particular, we study the free energy fluctuations when the temperature is in a window of size $n^{-1/3}\sqrt{\log n}$ around the critical temperature, the same window for the SSK model. Within this transitional window, the asymptotic fluctuations of the free energy are the sum of independent Gaussian and Tracy--Widom random variables. The talk is based on joint work with Elizabeth Collins-Woodfin.

Friday, November 10th Duncan Dauvergne
University of Toronto


Friday, December 1st Philippe Sosoe
Cornell University


Friday, December 8th Arjun Krishnan
University of Rochester


Stochastics Seminar for Spring 2022 is organized at the University of Utah by B. Cooper Boniece and Xiao Shen.
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