Abstract
AbstractWe study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range . In particular, we first focus on the expected time for a random walk to hit a given vertex or visit, that is, cover, all vertices. We show that, a.a.s. (with respect to the HRG), and up to multiplicative constants: the cover time is , the maximum hitting time is , and the average hitting time is . We then determine the expected time to commute between two given vertices a.a.s., up to a small factor polylogarithmic in , and under some mild hypothesis on the pair of vertices involved. Our results are proved by controlling effective resistances using the energy dissipated by carefully designed network flows associated to a tiling of the hyperbolic plane, on which we overlay a forest‐like structure.
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