Probability Archives for Fall 2023 to Spring 2024
Stationary solutions for 1D Burgers equations and KPZ scaling
When: Wed, September 28, 2022 - 1:00pmWhere: MATH 1310
Speaker: Liying Li (University of Michigan) -
Abstract: In the first part, we will talk about the stationary solutions for 1D stochastic Burgers equations and their ergodic properties. We will classify all the ergodic components, establish the ``one force---one solution'' principle, and obtain the inviscid limit. The key objects to study are the infinite geodesics and infinite-volume polymer measures in random environments, and the ergodic results have their counterparts in the geodesic/polymer language. In the second part, we will present a random point field model that is motivated by the coalescing and monotone structure of the optimal paths in random environments that arise in many KPZ models. The 2/3 transversal exponent from the KPZ scaling becomes a natural parameter for the renormalization action in this model, and can be potentially extended to values other than 2/3. Some preliminary results are given.
Understanding the upper tail behaviour of the KPZ equation via the tangent method
When: Wed, October 5, 2022 - 1:00pmWhere: MATH 1310
Speaker: Milind Hegde (Columbia University) - http://math.columbia.edu/~milind/
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.
Joint work with Shirshendu Ganguly.
Shift invariance of half space integrable models
When: Wed, October 12, 2022 - 1:00pmWhere: MATH 1310
Speaker: Jimmy He (MIT) - https://he-jimmy.github.io/
Abstract: I'll discuss recent work on shift invariance in a half space setting. These are non-trivial symmetries allowing certain observables of integrable models with a boundary to be shifted while preserving their joint distribution. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure, both in a half space setting. An application to the asymptotics of a half space analogue of the oriented swap process is also given.
Some recent development in the weak noise theory for the KPZ equation
When: Tue, February 21, 2023 - 3:30pmWhere: Kirwan Hall 1310
Speaker: Li-Cheng Tsai (University of Utah) - https://lc-tsai.github.io/
Abstract: The variational principle, or the least action principle, offers a framework for the study of the Large Deviation Principle (LDP) for a stochastic system. The KPZ equation is a stochastic PDE that is central to a class of random growth phenomena. In this talk, we will study the Freidlin--Wentzell LDP for the KPZ equation through the lens of the variational principle. Such an approach goes under the name of the weak noise theory in physics. We will explain how to extract various limits of the most probable shape of the KPZ equation in the setting of the Freidlin--Wentzell LDP. We will also review the recently discovered connection of the weak noise theory to integrable PDEs.
This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Yier Lin.
Superdiffusivity and localization of the CDRP
When: Wed, February 22, 2023 - 1:00pmWhere: MATH 3206
Speaker: Weitao Zhu (Columbia University) -
Abstract: The continuum directed random polymer (CDRP) model is a universal scaling limit of discrete direct polymers. In the long-time regime, the CDRP is expected to exhibit strong disorder polymer behaviors such as localization and superdiffusion. In this talk, we probe into the long-time geometry of the CDRP and demonstrate its pointwise localization and pathwise tightness under ⅔ KPZ scaling. In addition to confirming existing predictions about polymers under the continuous setting, we also explain how our techniques shed light on properties of the KPZ equation, such as ergodicity and limiting Bessel behaviors around the maximum. This talk is based on joint works with Sayan Das (Columbia).
Dickman distribution and the limit theorems on the Lee groups
When: Tue, February 28, 2023 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Stanislav Molchanov (UNC Charlotte) -
Abstract: Dickman distribution in the statistics of the [prime numbers were introduced in
1930 th by a German actuary K. Dickman at the completely non-rigorous level).
The justification of the corresponding mathematical conjecture goes mainly to
N.G. de Bruijn (in 1960 th ). This topic and the closely related results by V.
Goncharov on the cycles of the random permutations were extended by A.
Vershik (1970) and his students in several directions.
Recently, M. Penrose and A. Wade introduced another object under the same
name: Dickman distribution. They studied its application in the theory of random
graphs.
The talk will contain a review of old and new results in this area.
The stationary horizon as a universal object for KPZ models
When: Wed, April 5, 2023 - 1:00pmWhere: MATH 3206
Speaker: Evan Sorensen (UW Madison) - https://sites.google.com/wisc.edu/evan-sorensen
Abstract: The last 5-10 years has seen remarkable progress in constructing the central objects of the KPZ universality class, namely the KPZ fixed point and directed landscape. In this talk, I will discuss a third central object known as the stationary horizon (SH). The SH is a coupling of Brownian motions with drifts, indexed by the real line, and it describes the unique coupled invariant measures for the directed landscape. I will talk about how the SH appears as the scaling limit of several models, including Busemann processes in last-passage percolation and the TASEP speed process. I will also discuss how the SH helps to describe the collection of infinite geodesics in all directions for the directed landscape. Based on joint work with Timo Seppäläinen and Ofer Busani.
From the KPZ equation to the directed landscape
When: Wed, April 12, 2023 - 1:00pmWhere: MATH 3206
Speaker: Xuan Wu (University of Chicago) - https://math.uchicago.edu/~xuanw/
Abstract: This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.
Optimal CLT rates in transport distances under weak dependence
When: Wed, April 26, 2023 - 3:00pmWhere: MATH 3206
Speaker: Yeor Ross Hafuta (UMD) - https://sites.google.com/mail.huji.ac.il/yeor-hafouta
Abstract: The transport distance of order p (aka the Wasserstein metric W_p) measures the distance between two real valued random variables by means of the smallest possible L^p norm of their difference, after coupling.
An old problem from the late 50's and early 60's was to obtain optimal conditions for optimal CLT rates (aka Berry-Esseen theorems) in W_p for self-normalized partial sums S_n=X_1+...+X_n of iid random variables. The problem was solved by E. Rio (2009). Bobkov (2018) extended the optimal rates to independent but not identically distributed r.v. and, additionally, answered a problem by Rio about the optimal constants appearing in the Berry-Esseen theorem in W_p.
In the talk we will discuss extensions of the Berry-Esseen theorem in W_p for several classes of weakly dependent (not necessarily stationary) sequences of random variables like inhomogeneous Markov chains, products of random matrices and sequential expanding dynamical systems. All the results are new already in the stationary case.