In the following problem, we have an unfortunate thief. The thief undergoes a $1d$ simple symmetric random walk on the line starting from the origin at $t=0$.
At the same time, there are police officers that begin their own independent simple symmetric random walks all throughout the lattice - that is, there is one police officer on each lattice point $na, {n \in \mathbb{Z}^*}$ at $t=0$. For example, for $a=4$, there is a police officer on each of the sites $(...,-8,-4,4, 8,12,...)$. For uniformity of notation, call $\tau$ the first collision time between our walker and any police officer. Assume $a$ is even so that the police officers and thief can collide.
What is the asymptotic behavior of the probability $\mathbb{P}(\tau \geq t)$ that the thief survives for at least a time $t$ without any collisions with police officers? I'm also happy with upper bounds demonstrating that the tail probability is at least exponentially suppressed in $\frac{t}{a^2}$; i.e. $\mathbb{P}(\tau \geq t) \leq c_1 e^{-c_2 \frac{t}{a^2}}$ for some appropriate $c_1, c_2>0$.
I want to say my motivation for the exponential bounds. Suppose the officers were stationary and not random walking; the question is then about the first exit time $\tau$ from the region $(-a, a)$ starting at the origin. For a Brownian motion started at the origin, the first exit time from this region obeys $\mathbb{P}(\tau >t ) \leq c_1 e^{-c_2 \frac{t}{a^2}}$ for an appropriate pair of constants $c_1$ and $c_2$, as noted in an answer by Chris Jangjigian to another question. Allowing the police officers to themselves random walk gives our thief potentially a little bit of breathing room, but I suspect our thief's survival time will nevertheless suffer similar bounds.
EDIT: I ran some preliminary numerical checks for a similar model to the above. I find that the tail probability seems fairly well described by $\mathbb{P}(\tau \geq t) \approx c_1 e^{-c_2 \sqrt{\frac{t}{a^2}}}$ at large $t$. It's a lighter tail than any power-law, at least, but note that this is a heavier tail than I expected. I wonder if there's a nice heuristic way to see this stretched exponential behavior. As an aside, I'll note that in the case of only two officers, we had an even heavier tail of $\frac{1}{t^{3/2}}$.