Dear colleagues,

I would like to inform you of several online lectures in probability next week from Feb 21-23, including a short lecture series by Prof. Nam-Gyu Kang (KIAS), and culminating in the 2022 KMS Probability Workshop on Feb. 23:

Monday, Feb 21

 

9:30-11:00 Nam-Gyu Kang, KIAS

An introduction to conformally invariant random curves

Schramm introduced stochastic Loewner evolutions (SLEs) as the only possible candidates for the scaling limits of interface curves in several critical lattice models. His approach led to the rigorous proofs of some crucial conjectures in statistical physics. I will review some significant developments of SLE theory.

11:00-12:15 Greg Markowsky

Some recent results on planar Brownian motion: a Skorokhod embedding problem and fast exits from simply connected domains.

I will discuss some recent results on planar Brownian motion, including the following analogue of the classical Skorohod embedding problem. If µ is a distribution on the real line, is there a planar domain U for which the real part of a stopped planar Brownian motion upon exiting U has the distribution µ? It turns out that this problem is solvable in several different ways under certain assumptions. I will also discuss another problem, namely conditions under which a Brownian motion is likely to exit a simply connected domain in a very short time. These results are based on joint work with Maher Boudabra and Dimitrios Betsakos. 

Tuesday, Feb 22

 

9:30-11:00 Nam-Gyu Kang, KIAS

A mathematical guide to conformal field theory

I will give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. I will accurately define and explain some basic concepts in conformal field theory, such as Ward's identities, stress tensor, and vertex fields in terms of correlation functions of statistical fields.

 

11:00-12:15 Michael Damron

Translation-invariant nearest neighbor graphs

Abstract: Given edge-lengths (t_e) assigned to the edges of Z^d for d geq 2, each vertex draws a directed edge to its closest neighbor to form the "nearest neighbor'' graph. Nanda-Newman studied these graphs when the t_e's are i.i.d. and continuously distributed and showed that the undirected version has only finite components, with size distribution whose tail decays like 1/n!. I will discuss recent work with B. Bock and J. Hanson in which the t_e's are only assumed to be translation-invariant and distinct. Here, we can show that there are no doubly-infinite paths and completely characterize the set of possible graphs. In particular, for d=2, the number of infinite components is either 0,1, or 2, and for d geq 3, it can be any nonnegative integer. I will also mention relations to both geodesic graphs from first-passage percolation and the coalescing walk model of Chaika-Krishnan.

Wednesday, Feb 23 (KMSA Probability Workshop)

 9:30-11:00 Nam-Gyu Kang, KIAS

Conformal field theory for multiple SLEs and its classical limit

Multiple SLEs describe several random interfaces consistent with conformal symmetries. I will explain a version of conformal field theory with the insertion of N-leg operators with screening to show that this version produces a collection of martingale-observables for multiple SLEs. These observables become integral motions for multiple SLE(0) curves as the SLE parameter tends to 0. Using this method, I will explain Peltola-Wang's description of these non-random curves as the real locus of the rational functions with prescribed critical points.

Based on joint work with Tom Alberts (Utah) and Nikolai Makarov (Caltech).

11:00-12:15 Panki Kim, Seoul National University

Non-local Operators Whose Kernels Degenerate at the Boundary

In this talk, we discuss the potential theory of Markov processes with jump kernels degenerating at the boundary of the half space. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of parameters. Depending on the regions where parameters belong, the estimates on the Green functions are different. As applications, we completely determine the region of the parameters where the boundary Harnack principle holds or not. This talk is based on joint works with Renming Song and Zoran Vondraček.

2:00-3:15 Hyun Jae Yoo, Hankyong National University

Central limit theorem and mixture of Gaussians in the limit of open quantum walks

Abstract:  Quantum walks, e.g., Hadamard walks, are known to be more efficient in some quantum algorithms, such as search algorithm, than the classical walks. Open quantum walks introduced by Attal, et al, (JSP, 2012) generalizes the classical random walks in one hand and on the other hand it is proposed to implement the dissipation and decoherence, which is inevitable subject to an interaction with the environment, in the description of quantum walks. In this talk we discuss the central limit theorem for the open quantum walks on the integer lattices, owing to the work by Attal, et al (Ann. Henri Poincare, 2015). Then we extend the result to the crystal lattice models as well as show that the mixture of Gaussians will come out in general. It is based on the joint works with Chul Ki Ko.  

 3:15-4:30 Kunwoo Kim, POSTECH

Oscillation and decay of the solution to stochastic heat equations

In this talk, we consider a nonlinear stochastic heat equation driven by space-time white noise on the interval with periodic boundary condition and positive initial data. If the noise is multiplicative (i.e., our equation is the parabolic Anderson model), the almost sure Lyapunov exponent at a fixed spatial point is known. In this talk, we show a stronger result that says that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. It follows that, for the parabolic Anderson model, almost sure Lyapunov exponents of supremum and infimum of the solution are the same, which implies that the entire path decays almost surely at an exponential rate. This is based on joint work with Davar Khoshnevisan and Carl Mueller.