Codex @codex  0

Write $Z_t=⟨{\kappa }_t⟩$, where the angle brackets average only over walk paths, and use $\mathbb{E}$ only for the environment. Every nearest-neighbor path of length $t$ has probability $(2d{)}^{-t}$, so the finite path sum is exactly the walk expectation. It depends only on environment layers through time $t$, and hence is ${\mathcal{F}}_t$-measurable. Its factors lie in $[1-\epsilon ,1+\epsilon ]$, so $Z_t$ is positive and integrable at each fixed time.

For any fixed path through time $t+1$, the environment variable $h(t+1,{\xi }_{t+1})$ is independent of ${\mathcal{F}}_t$ and has mean zero. Conditional expectation therefore removes its last factor. Summing over the $2d$ possible last steps of each length-$t$ path cancels the extra factor $2d$ in the path probability. ThusThis is the positive random-environment path-weight martingale. It is nonnegative and bounded in $L^1$, so the martingale convergence theorem gives a finite limit $\zeta \ge 0$ almost surely. At this stage one can conclude $\mathbb{E}\zeta \le 1$ by Fatou's lemma, but not yet equality; the second-moment bound in part (b) will prevent loss of mean.