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 @math is equivalent, up to universal constants, to the square of the expected maximum of the Gaussian free field on @math , scaled by the number of edges in @math . 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 @math for @math -vertex graphs, due to Kahn, Kim, Lovasz, and Vu (2000).
Furthermore, for a few families of specific examples, the asymptotics of the cover time have been calculated more precisely. These include the work of Aldous @cite_37 for regular trees, Dembo, Peres, Rosen, and Zeitouni @cite_22 for the 2-dimensional discrete torus, and Cooper and Frieze @cite_45 for the giant component of various random graphs.
@cite_37 @cite_45 @cite_22
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We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length @math . Dembo, Peres, Rosen, and Zeitouni [Ann. Math. 160:433-464, 2004] identified the leading term in the asymptotics for the cover time as @math goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler [Probab. Theory Relat Fields. 167:461-552, 2017]. We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus,and show that in both cases with probability approaching @math as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progresstowards a conjecture of Bramson and Zeitouni (2009). We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009). Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r â¥ 3 is fixed and n = |G| â ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the 'scaling window' of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobas and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.