Can One Condition a Killed Random Walk to Survive?

We consider the simple random walk on \(\mathbb {Z}^d\) killed with probability p(|x|) at site x for a function p decaying at infinity. Due to recurrence in dimension \(d=2\), the killed random walk (KRW) dies almost surely if p is positive, while in dimension \(d \ge 3\) it is known that the KRW dies almost surely if and only if \(\int _0^{\infty }rp(r)dr = \infty \), under mild technical assumptions on p. In this paper we consider, for any \(d \ge 2\), functions p for which the random walk will die almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion \((\Lambda _R)_{R \in \mathbb {N}}\) of \(\mathbb {Z}^d\), does the KRW conditioned to leave \(\Lambda _R\) before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for \(p(r) = o(r^{-2})\), and that it is not for \(p(r) = \min (1, r^{-\alpha })\) for \(\alpha \in (14/9,2)\). This question is connected to branching random walks and the infinite snake. More precisely, in dimension \(d=4\), the infinite snake is related to the KRW with \(p(r) \asymp (r^2\log (r))^{-1}\), therefore our results imply that the infinite snake conditioned to avoid the origin in four dimensions is well-defined.