We exhibit a strong connection between cover times of graphs, Gaussian processes, and Talagrand's theory of majorizing measures. In particular, we show that the cover time of any graph G is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on G, scaled by the number of edges in G. This allows us to resolve a number of open questions. We give a deterministic polynomial-time algorithm that computes the cover time to within an O(1) factor for any graph, answering a question of Aldous and Fill (1994). We also positively resolve the blanket time conjectures of Winkler and Zuckerman (1996) , showing that for any graph, the blanket and cover times are within an O(1) factor. The best previous approximation factor for both these problems was O((log log n) 2 ) for n-vertex graphs, due to Kahn, Kim, Lovász, and Vu (2000) .
More recently, a breakthrough was made by Ding, Lee, and Peres REF whereby the cover time was given (up to a constant factor) in terms of the Gaussian free field.
Cover times, blanket times, and majorizing measures
STOC '11
Mathematics Computer Science