Lemma: Escape bound and non-explosion

Fix $s>2\lambda$ and put

For $L\ge 1$ and $t\ge 0$,

Consequently $I(t)$ is finite almost surely for every finite $t$.

Proof. Any path reaching $\partial {\Lambda }_L$ has, after loop erasure, at least $L$ edges. There are at most $4\cdot 3^{k-1}$ self-avoiding paths of length $k$ starting at $0$. For each such path, independence and Markov's inequality give

Sum over $k\ge L$. The resulting bound tends to zero as $L\to \infty$. An infinite $I(t)$ would produce paths reaching arbitrarily large boundaries with cost at most $t$. This event has probability zero. Take a countable intersection over integer $t$.