**Dates & Times:**

- Di. 23.10, 16-18: Incidence matrix \(B\), Laplacian matrix \(L=BB^t\), spanning trees, Kirchhoff's matrix-tree theorem and its proof (using edge contraction and deletion): \(\tau(G) = \det(L_G[i]) = n^{-1}\prod_{i=2}^n \lambda_i\). Uniform spanning trees (USTs). Statement of the Aldous-Broder algorithm.
- Mi. 24.10, 10-12: One-edge probabilities \({\bf P}(e\in \mathcal{T})\) and random walk interpretation. Multi-edge probabilities \({\bf P}(e,f\in \mathcal{T})\) and negative correlations. A hint of determinantal point processes. Statement of the transfer-current matrix theorem of Burton & Pemantle.
- Fr. 26.10, 9-11: Doubly stochastic irreducible Markov chains have uniform invariant measure. Proof that the Aldous-Broder algorithm generates UST (Key: check that the backward forests generated by SRW form an irreducible doubly-stochastic MC). Consequence: if \(e=\overrightarrow{xy}\), then \({\bf P}(e\in \mathcal{T})=\) probability that RW started at \(x\) uses the edge \(e\) to first hit \(y\). Electric network interpretation gives \({\bf P}(e\in \mathcal{T}) = R_{\rm eff}(x,y)\). Definition of the transfer-current matrix \(H(e,f)\): \(H(\overrightarrow{xy},\overrightarrow{zw}) = \phi_{\overrightarrow{xy}}(z) - \phi_{\overrightarrow{xy}}(w)\) where \(\Delta\phi_{\overrightarrow{xy}}=\delta_x-\delta_y\). \(H(\overrightarrow{xy},\overrightarrow{xy}) = R_{\rm {eff}}(x,y)\), hence proving the one-edge case of the Burton-Pemantle theorem. A preview of the two-edge case \({\bf P}(e,f\in \mathcal{T})\).
- Di. 6.11, 16-18: Edge contraction map \(\rho\) from \(G\) to \(G/f\). \(\bar\phi =\) lift of \(\phi\) from \(G/f\) to \(G\). Maximum principle for the harmonic function \(\bar\phi\). Expressing the two-edge probability as a determinant. Induction on the number of edges completes the proof of the Burton-Pemantle theorem: \({\bf P}(e_1, \cdots, e_r \in \mathcal{T}) = \det(H(e_i, e_j))_{1\leq i,j\leq r}\). Probability of other cylinder events: e.g. \({\bf P}(e_1, \cdots, e_r \notin \mathcal{T}) = \det((I-H)(e_i, e_j))_{1\leq i,j\leq r}\).
- Mi. 7.11, 10-12: The abelian sandpile model. The sandpile group on a finite sinked graph: \(\mathbb{Z}^V/\mathbb{Z}^V\Delta'\), which is in 1-to-1 correspondence with the recurrent sandpile states. Dhar's multiplication by identity test (for a "live" application see this paper). The burning bijection between spanning trees and recurrent sandpiles (Majumdar-Dhar). Random walks on sandpiles = USTs. Epilogue.

- Robin Pemantle's survey "Uniform Random Spanning Trees."
- Russell Lyons and Yuval Peres, "Probability on Trees & Networks". Mainly Chapters 2 & 4.
- Burton and Pemantle's 1993 paper in
*Ann. Probab.* - Animation of Wilson's algorithm to generate USTs on the square lattice (Mike Bostock)
- Lectures on the abelian sandpile model: Frank Redig (2005); Antal Járai (2014).

**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.

- 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.

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