18.177 Stochastic Processes in Physics.
UNIFORM SPANNING TREE TOPICS (from the online book by Lyons and Peres): reversible Markov chains, stationary distribution, harmonic functions on graphs, maximum principle solution to the Dirichlet problem as the expected value at the time the processes hits the boundary (proved on graphs, stated for Brownian motion in the plane), electrical networks, probabilistic interpretations of effective resistance, star space and cycle space, effective resistance as the minimum energy of a unit flow, Rayleigh monotonicity, Kirchhoff's theorem relating edge probabilities in the UST to the unit current flow, Foster's theorem as a corollary, negative correlations of edges in the UST (a consequence of Kirchhoff's thm and Rayleigh), Burton-Pemantle theorem expressing edge marginals as determinants (note: single-marginal case is Kirchoff's theorem, two-edge case gives negative correlations), Wilson's algorithm, and cycle popping PROBLEM SET ONE (DUE MARCH 2) PERCOLATION TOPICS (from Percolation by Grim