A simple characterization of the effective resistance metric on vertex transitive graphs

The effective resistance satisfies the triangle inequality and thus defines a metric. For finite graphs, this metric contains all of the information about the expected hitting times between pairs of vertices as well as about the cover time of the graph (= the first time by which every vertex has been visited at least once). We show that for transitive graphs, the effective resistance between a pair of vertices o and x, which are at distance r from one another, is comparable (up to a constant multiplicative factor, depending only on the degree) to the expected number of returns to the origin by time r^2. We use this (in the transitive bounded degree setup) to: (1) Give a complete characterization of the effective resistance metric up to quasi-isometries, with effective O(1) implicit constants. (2) Give simple formulas, involving few natural geometric quantities, for the orders of the expected cover time and the maximal (over all pairs) effective resistance between a pair of vertices. Joint work with Lucas Teyssier (UBC) and Matt Tointon (U. Bristol).