## 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.

Outline of plans: (version 0.5, still subject to change)
• Overview of the hydrodynamic program on (non-)Euclidean settings: making connections with random walks on graphs. Path picking method of Guo-Papanicolaou-Varadhan and Diaconis-Saloff-Coste. Why this does not give optimal results on non-translationally-invariant state spaces (congestion ratio vs. Poincare constant).
• Spectral inequalities in particle systems, via projection of many-particle system to a single particle. The interchange process. Aldous' spectral gap conjecture, the octopus inequality of Caputo-Liggett-Richthammer, Chen's moving particle lemma. Analogs for the zero-range process (Hermon-Salez, Chen).
• Dirichlet forms (for single-particle diffusion, for particle systems, and their connections). Feynman-Kac formula. Kipnis-Varadhan inequality. Estimating the spectral gap or the $$H_{-1}$$ norm of additive functionals.
• Replacement of microscopic functionals by coarse-grained macroscopic ones. Local ergodic theorem on resistance spaces (1-block + 2-blocks estimates). The 2nd-order Boltzmann-Gibbs principle (for equilibrium fluctuations in the exclusion process, a new proof bypassing multiscale analysis by Chen).
• Convergence of stochastic processes: Aldous, Mitoma, Prohorov-Kolmogorov-Centsov. Hydrodynamic limit theorems on resistance spaces. Density fluctuations in WASEP and their scaling limits: Edwards-Wilkinson scaling regime (Ornstein-Uhlenbeck process), Kardar-Parisi-Zhang scaling regime (stochastic Burgers' equation).
• Boundary-driven exclusion process on the Sierpinski gasket: LLN, LDP. Connections with macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim.
Acknowledgement. I would like to thank Prof. Patricia Gonçalves for the kind invitation to deliver this minicourse.
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