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For the supercritical contact process on the hyper-cubic lattice started froma single infection at the origin and conditioned on survival, we establish twouniformity results for the hitting times $t(x)$, defined for each site $x$ asthe first time at which it becomes infected. First, the family of randomvariables $(t(x)-t(y))/|x-y|$, indexed by $x \neq y$ in $\mathbb{Z}^d$, isstochastically tight. Second, for each $\varepsilon >0$ there exists $x$ suchthat, for infinitely many integers $n$, $t(nx) < t((n+1)x)$ with probabilitylarger than $1-\varepsilon$. A key ingredient in our proofs is a tightnessresult concerning the essential hitting times of the supercritical contactprocess introduced by Garet and Marchand (Ann.\ Appl.\ Probab., 2012).

Uniformity of hitting times of the contact process