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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.

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  • $\begingroup$ I forget to mention that the asymptotic is in m for some fixed n (large). $\endgroup$
    – seb
    Mar 18, 2014 at 11:14

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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 $m\to+\infty$.

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