Random walk inside a square (probability of escape before returning)

Question

My question comes from Exercise 9.7. of the book "Markov Chains and Mixing Times (2nd edition)" written by David A.Levin and Yuval Peres. Specifically, let $B_n$ be the subset of $\mathbb{Z}^2$ in the box of side length $2n$ centered at $0$. Let $\partial B_n$ be the set of vertices along the perimeter of the box. The problem statement asks us to show that for simple random walk on $B_n$, $$\lim_{n\to \infty} \mathbb{P}_0\{\tau_{\partial B_n} < \tau^+_0\} = 0.$$ I think it is intuitively clear but have no idea on how this can be justified analytically. Since this problem is in Chapter 9, I guess the author wants us to use the theory of network reduction laws/rules developed in the context of "random walks on networks". Any hint or help are greatly appreciated!

Answer 1

I think I have something, if I dindn't make any mistake.

$\mathbb{P}_0\{\tau_{\partial B_n} < \tau^+_0\} = \sum_{k \in \mathbb{N} } \mathbb{P}\{\tau_{\partial B_n} < \tau^+_0 |\tau^+_0 =k \} \mathbb{P}\{ \tau^+_0 =k \} = \sum_{k \geq n } \mathbb{P}\{\tau_{\partial B_n} < \tau^+_0 |\tau^+_0 =k \} \mathbb{P}\{ \tau^+_0 =k \} \leq \sum_{k \geq n } \mathbb{P}\{ \tau^+_0 =k \} = \mathbb{P}\{ \tau^+_0 \geq n \}$

Overall $\mathbb{P}_0\{\tau_{\partial B_n} < \tau^+_0\} \leq \mathbb{P}\{ \tau^+_0 \geq n \}$ which mean that if a random walk touch the boundary of $B_n$ and then go back to $0$ then the first return map to $0$ take at least $n$ step.

Obviously $\mathbb{P}\{ \tau^+_0 \geq n \} \underset{n \to \infty}{\to} 0$