Consider a reversible random walk on (say) $\mathbb{Z}$, are there any estimate for the following 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 decrease exponentially in distance, but the Simple random walk case can be already helpful for me, thanks.