**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.

- 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 Dynkinâ€™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.

- Caputo, Liggett, and Richthammer, "Proof of Aldous' spectral gap conjecture." JAMS (2010).
- C., "The moving particle lemma for the exclusion process on a weighted graph." ECP (2017).
- Doyle and Snell, "Random walks and electric networks." 2006 GNU version.
- Kumagai, "Random walks on disordered media and their scaling limits." St. Flour 2010 Lecture Notes.
- Gonçalves, Jara, and Simon, "Second order Boltzmann-Gibbs principle for polynomial functions and applications." J. Stat. Phys. (2017).
- Gonçalves and Jara, "Nonlinear fluctuations of weakly asymmetric interacting particle systems." Arch. Rational Mech. Anal. (2014).

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