Asymptotic estimate of random walk involved hitting time and return time

Question

Consider a reversible random walk on (say) $\mathbb{Z}$. Are there any estimates for the probability $\mathbb{P}(\tau_n=m<\tau_0^+)$ where $\tau_n$ is the first hitting time at site n and $\tau_0^+$ is the first return time to the starting point 0?

The conductance decreases exponentially in distance, but the Simple random walk case can be already helpful for me.

Answer 1

I know for a fact that when $0\leq a\leq b$, we have $\mathbb{P}_a[\tau_0 < \tau_b] = 1-\tfrac{a}{b}$. So my guess is that your probability is equivalent to $\tfrac{1}{m}$ as $m\to+\infty$.