Joe P. Chen's Research Lectures (Autumn 2018 & Winter 2019)

Bielefeld (Autumn 2018) | Lisboa (Winter 2019)

Universität Bielefeld Luftaufnahme  

Bielefeld-Seoul IRTG 2235 Scientific Block Course: Random processes induced by Laplacian determinants

Dates & Times:

Location: D5-153.
Office hours: By appointment in V3-222.
Plans: We will discuss several concrete examples of spatial random processes whose moments are encoded by Laplacian determinants on a d-regular (or a weighted) graph. Examples include: uniform spanning trees, cycle-rooted spanning forests, recurrent sandpiles. Points of emphasis will be on the bijections between the said models, scaling limits, and sampling algorithms (Aldous-Broder, Wilson).
Some references:

IST Lisboa Minicourse: Hydrodynamic limit of particle systems on resistance spaces

Dates: January 25, 2019 through February 6, 2019. Mon 15:30-17:00, Wed 15:30-17:00, Fri 14:00-15:30.
Location: Room 4.35, Mathematics Department.
Official announcement from Tecnico Lisboa, Minicourse schedule
Minicourse description:

A main topic in probability theory is the study of scaling limits of random processes. One class of problems deals with scaling limits of single-particle Markov processes to a diffusion process. Another class of problems deals with scaling limits of many-particle Markov processes to a deterministic or stochastic differential equation. The former class has been studied on many state spaces, such as Euclidean spaces, manifolds, graphs, groups, etc. The latter class has been studied on Euclidean spaces, but not as much on non-Euclidean spaces.

The goal of my mini-course is to describe my recent progress on establishing scaling limits of many-particle systems on state spaces which are bounded in the resistance metric, a.k.a. "resistance spaces." These include trees, fractals, and random graphs arising from critical percolation. As a concrete example, we can establish scaling limits of the weakly asymmetric exclusion process on the Sierpinski gasket interacting with 3 boundary reservoirs, which generalizes (in a nontrivial way) the analysis on the unit interval interacting with 2 boundary reservoirs. I will explain the key ideas behind these results, and discuss connections to the analysis of (S)PDEs, and issues of non-equilibrium statistical physics, on resistance spaces. From a technical point of view, I will address some novel functional inequalities for the exclusion process that relates to electrical resistance, and describe how they are used to effect "coarse-graining" in passing to the scaling limits.

Outline of plans: (version 0.5, still subject to change) Acknowledgement. I would like to thank Prof. Patricia Gonçalves for the kind invitation to deliver this minicourse.
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