# Random walk on a graph

For a random walk say from point $x$ to $y$ on a graph,

How is the probability of a Random walker reaching point $y$ before returning to $x$ related to the expected of the number of visits to point $x$, $E(T_{x})$ and the expected number of visits to y, $E(T_{y})$ after infinitely many steps? (If there does exist one can you please prove it)

• with classical notations, $ET_x$ is the average time to return to $x$ starting from $x$. – mookid Mar 26 '14 at 4:35
• yes that's correct – user42382 Mar 26 '14 at 4:42
• so, not the expected number of visits. – mookid Mar 26 '14 at 4:48
• well i was kinda assuming they're the same – user42382 Mar 26 '14 at 5:03
• isn't the number of visits infinite? – mookid Mar 26 '14 at 5:03