The regular meeting time is Mondays 2:30-3:25 in Seminar Room, 4206 French Hall. for more info.
Schedule
Week of Jan 23: Monday 2:30-3:25, Jaime Garza: Binary Sequential Representation for the Two Parameter Chinese Restuarant Process
In this seminar I will be talking about random set partitions. In particular, the random set partitions that are generated by the Chinese Restaurant Process (CRP). This process builds sequences of set partitions $(\Pi_n)$ such that each $\Pi_n$ is a set partition of $\{1,2,\dots,n \}$ and $\Pi_{n+1}$ is built from $\Pi_n$ by assigning $n+1$ to an existing block of $\Pi_n$ or creating a new singleton block with $n+1$. Most of the talk will be centered around what is known as the two parameter CRP for which many properties are known. I will also be presenting results from a paper by James Young titled "Binary Sequential Representations of Random Partitions" which describes a way in which binary sequences can be used to build random set partitions with distribution following that of the two parameter CRP.
Week of Jan 30: Monday 2:30-3:25: Xiaoqin Guo:
Optimal convergence rates in stochastic homogenization in a balanced random environment
Stochastic homogenization studies the convergence of differential equations with random coefficients to a deterministic "effective" equation. A closely related problem is the quenched central limit theorem (QCLT) for random walks in random environment (RWRE) which states that the RWRE converges to a Brownian motion with deterministic diffusivity in large scale. In this talk we consider the stochastic homogenization of non-divergence form equations on the integer lattice and the corresponding model of random walk in a i.i.d. random environment (which is a martingale). For deterministic PDE it is known that the invariant measure of this equation does not have explicit expressions and only admits $(d+1)/d$-th moment.
We will discuss the correlation structure of the invariant measure in the random environment and show a quantitative law of large numbers. We will also derive optimal quantitative estimates for rates of the QCLT by quantifying the ergodicity of the environment process. Joint work with Hung V. Tran (UW-Madison).
Wednesday, Feb 1, 3:35-4:20 (Graduate Seminar) French Hall 4221: Wlodek Bryc,
Characterizations of laws and processes by their symmetries: The case of the normal law
I will present a very intuitive characterization of the normal law discovered by Hershel in 1850. Then I will state some "generalizations": Polya's theorem (1923), Bernstein theorem (1941), and point out that they have consequences of interest for statistics.
Week of Feb 6: No activities in the seminar. (This is a week with seven job talks.)
Week of Feb 13: Monday 2:30-3:25, Jacek Wesolowski (Warsaw, Poland), The Matsumoto-Yor property - a revisit after 20+ years
I will review results related to an independence (MY) property of the GIG (generalized inverse Gaussian) and Gamma distributions, which was discovered by H. Matsumoto and M. Yor in the very end of the last century (their paper appeared only in 2001). This will be a (lengthy) introduction to recent developments related to the generalized MY property described by Croydon and Sasada in 2020 in their search for stationary measures of random iterations on $\mathbb{Z}^2$. In particular, I will present a new characterization of the GIG distribution as well as the generalized MY property in the cone of positive definite matrices. This results come from a recent paper joint with G. Letac (Toulouse, France).
Week of Feb 20:
Monday, Feb 20: 2:30-3:25PM, Jacek Wesolowski The Matsumoto-Yor property - a revisit after 20+ years (continued).
Tue, Feb 21: 11:15-12:25 ZOOM talk by Konstantin Matetski (MSU)
Polynuclear growth and its exact solution"
venue: Probab. Seminar in Warsaw (online) (ask Wlodek for ZOOM details)
Polynuclear growth (PNG) is a Markov process on a space of piece-wise constant functions and is one of the simplest models describing crystal growth. PNG has also occurred in other seemingly unrelated problems. For example, when started from a particular initial state, its one-point value equals the length of the longest increasing subsequence for uniformly random permutations (whose asymptotic behavior was first studied by S. Ulam).
In my joint work with J. Quastel and D. Remenik, the distribution functions of PNG with arbitrary initial conditions were computed for the first time. The exact formulas allowed us to study the scaling limits of the model and to establish its convergence to the KPZ fixed point. Moreover, these formulas allowed us to express the distribution function of PNG in terms of the solutions of the Toda lattice, one of the classical integrable systems. Such a relation between random growth models and classical integrable systems is still poorly understood.
Week of Feb 27: Monday 2:30-3:25, Yier Lin (U. Chicago) A lower-tail limit of the KPZ equation
The Kardar-Parisi-Zhang (KPZ) equation is an important model for the random interface growth. In this talk, I will explain how to extract the lower tail limit of the most probable shape of the KPZ equation in the setting of the Freidlin-Wentzell LDP. The talk is based on joint work with Li-Cheng Tsai.
Week of March 6: Monday 2:30-3:25, Xuan Wu (U. Chicago), From the KPZ equation to the directed landscape
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.
Week of March 13: (spring break)
Week of March 20:
Monday 2:30-3:25: Florence Merlevede (Paris, France): Quadratic transportation cost in the central limit theorem for dependent sequences
In this talk, we give estimates of the quadratic transportation cost in the central limit theorem for a large class of dependent sequences. Applications to irreducible Markov chains or to dynamical systems generated by intermittent maps will be provided. This talk is based on a joint work with J. Dedecker and E. Rio.
Taft Lecture: Friday, March 24th from 4-5pm in Room 800 Swift Hall.
Anton Lukyanenko, Fractal Geometry of Numbers. (Refreshments in Faculty Lounge (Room 4118 French Hall) from 3:15-3:45pm)
Week of March 27: Note change of day!
Wednesday March 29 2:30 - 3:25, Seminar Room 4206 French Hall: Zongrui Yang (Columbia U): Stationary measure for six-vertex model on a strip
In this talk we will study a family of Markov chains governed by the six-vertex model on a strip. Their stationary measure can be solved by the matrix product ansatz and then characterized by the Askey-Wilson process. We study the limit of mean particle density and obtain the phase diagram.
Week of April 3: Monday 2:30-3:25 Guillaume Barraquand (Paris, France), Extreme statistics of diffusion in random media and the KPZ equation
In this talk, we will discuss the effect of disorder on the
extreme statistics of simple random walks and diffusion processes. In
particular, we will see that in presence of a disordered environment,
the large deviations of one dimensional random walks has similar
fluctuations as the largest eigenvalues of Hermitian random matrices.
This is based on the detailed study of an exactly solvable model, the
Beta random walk. More generally, we will show how models of diffusion
in random media are connected to the Kardar-Parisi-Zhang equation and
universality class. If time permits, we will also discuss about the
stationary measures of the Beta Random walk, and models with boundaries.
Week of April 10: Monday 2:30-3:25: Alexey Kuznetsov (Toronto, Canada), The double gamma function and its applications in probability (and beyond)
A long long time ago (in the late 19th century, to be more precise) Ernest William Barnes introduced the double gamma function. It wasn't used much for almost a hundred years, but recently it started to appear more and more frequently in solutions to various problems in probability, number theory and in other areas. Despite these appearances, the function is still considered rather "exotic" and it is not widely known. The main goal of this talk is to demystify the double gamma function and to try to convince you that it is in fact a very nice, simple and useful special function. I will highlight the many parallels that this function has with the classical gamma function and with elliptic functions, then I will discuss various applications of the double gamma function in probability (and beyond), and, finally, I will explain how to compute this function very efficiently and to a high precision.
Sat, April 15: AMS meeting Probability-related sessions:
