Theorem: Transfer of an asymptotically identical passage front

Let $S_n(x)$ be nonnegative random arrival times, coupled to $T$. For every fixed $R$, suppose there are $b_n(R)=o(n)$ and ${ϵ}_n(R)\to 0$ such that

Then for $t_0>0$ and $\delta >0$, with probability tending to one,

and

within any fixed observation box $|x{|}_2\le Rn$. If the rescaled arrival sets are also contained in one fixed compact set with probability tending to one, their filled versions converge in Hausdorff distance in probability to $t_0\mathcal{B}$.

For every fixed $a\in (0,1)$, the corresponding annealed level sets

have the same inner and outer macroscopic inclusions, under the same uniform compact-containment condition.

Proof. On the comparison event and the event in (eq:uniform-shape), the sum of the two uniform errors is less than $\delta n$ for sufficiently large $n$. If $\mu (x/n)\le t_0-\delta$, this implies $S_n(x)\le n{t}_0$. If $\mu (x/n)\ge t_0+\delta$, it implies $S_n(x)>n{t}_0$. This proves the random-set inclusions.

Compact containment allows one observation box to contain the entire set with high probability. The two norm balls approach $t_0\mathcal{B}$ in Hausdorff distance as $\delta \downarrow 0$, and lattice-cell errors are $O(n^{-1})$.

The probability of the common good event tends to one. Uniformly over the inner set the arrival probability therefore tends to one; uniformly over the outer complement it tends to zero. Eventually these bounds lie on the appropriate sides of $a$.