The established shape input

The classical shape theorem supplies a deterministic norm $\mu$ such that, almost surely, for every $\epsilon >0$ and sufficiently large $t$,

The harmless cell convention is included in this statement CoxDurrett,ADH.

We shall also use its uniform passage-time consequence: for every fixed $R<\infty$,

To see this consequence, apply the shape inclusions at all sufficiently large passage times. They compare $T(0,x)$ and $\mu (x)$ multiplicatively outside the finite set infected before a fixed threshold. The remaining finite set contributes $o(n)$; then let $\epsilon \downarrow 0$.