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Heydenreich, Markus; Hirsch, Christian and Valesin, Daniel (2018): Uniformity of hitting times of the contact process. In: Alea-Latin American Journal of Probability and Mathematical Statistics, Vol. 15, No. 1: pp. 233-245

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For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times t(x), defined for each site x as the first time at which it becomes infected. First, the family of random variables (t(x) - t(y))/vertical bar x-y vertical bar, indexed by x not equal y in Z(d) , is stochastically tight. Second, for each epsilon > 0 there exists x such that, for infinitely many integers n, t(nx) < t((n + 1)x) with probability larger than 1-epsilon. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical contact process introduced by Garet and Marchanc (2012).

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