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

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.

Lecture plans:
• F 1/25: Random walks on graphs vs. exclusion process on graphs. Laplacian, invariant measure, Dirichlet energy, harmonic functions, spectral gap (and its significance). The Dirichlet-Thomson principle for random walks. Aldous' spectral gap conjecture for the interchange process.
• M 1/28: Hierarchy of stochastic processes on a fixed state space: Interchange process -> exclusion process -> random walk process. Electric networks: equivalent circuits (series law, parallel law, Delta-Y transform). A general formula for conductances under one-point network reduction (statement, no proof--uses Schur complements). Dirichlet energy is monotone decreasing under electric network reduction: classical result for random walks, but nontrivial for the interchange/exclusion process---the octopus inequality of Caputo-Liggett-Richthammer. Energy monotonicity implies the Dirichlet-Thomson principle (for random walks), resp. the moving particle lemma (for the interchange/exclusion process). Application #1 of moving particle lemma: local 1-block estimate for equilibrium density fluctuation (via Kipnis-Varadhan inequality, gets a cost which scales linearly with the diameter of the block in the resistance metric).
• W 1/30: Going from the local 1-block estimate to the local 2nd-order Boltzmann-Gibbs principle, sans multiscale analysis. A preview of the global 2nd-order Boltzmann-Gibbs principle on resistance spaces. Why 2BG? Coming from the quadratic field term in writing down Dynkin’s formula for the equilibrium density fluctuation field for ASEP.
• F 2/1: ASEP generator applied to the fluctuation field. Analysis of the Laplacian term (symmetric rate contribution), the martingale term, and the nonlinear term (antisymmetric rate contribution when $$\rho=1/2$$).
• M 2/4: Analysis on fractals: construction of the Dirichlet energy and the Laplacian on the unit interval, followed by the corresponding construction on the Sierpinski gasket. Remarks on the operator convergence. Putting everything back into Dynkin's formula for the density fluctuation field in ASEP.
• R 2/7 (3-4:30p, Rm 5.18): A Markov chain computation showing the time scale 5 on SG. Subcritical (Ornstein-Uhlenbeck) vs. critical scaling ("KPZ") in the nonlinear term, on the unit interval and on the Sierpinski gasket. A candidate for the stochastic Burgers' equation on SG, derived from ASEP.
Some references: Acknowledgement. I would like to thank Prof. Patricia Gonçalves for the kind invitation to deliver this minicourse.
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