2 Application of Path Coupling to Graph Colorings

Setup Given a Markov chain and pre-metric on Ω, our goal is to design a good coupling. Recall that a pre-metric on Ω is a weighted graph (Ω, Epre) in which every edge is a shortest path, and we name the induced shortest path metric d. In the previous lecture we proved the Path Coupling Theorem: Theorem 1 (Path Coupling). Suppose we have a coupling for all (x, y) ∈ Epre and suppose that there exists some α ∈ [0, 1] such that for all (x, y) ∈ Epre we have E [d(X ′, Y ′)|X = x, Y = y] ≤ (1− α)d(x, y). (*)