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Sep 13, 2010 at 12:15 comment added sleepless in beantown @Robin Chapman, you can effectively "touch the diagonal" by wrapping around, or having the biased random walk also reach a distance of n from the origin. So consider it as a biased random walk with the end condition being either returning to the origin or getting to the distance $n$.
Sep 13, 2010 at 12:03 comment added Seb67 Yes absolutely ! I completely missed this analogy, but now I understand much better what's going on.
Sep 13, 2010 at 11:59 comment added Robin Chapman That's more-or-less the same problem as considering the first expected return of an asymmetric one-dimensional random walk to the origin.
Sep 13, 2010 at 11:53 comment added Seb67 Yes of course the expected number of crossing is easy to compute. But can you say anything about the expected time you have to wait for the first crossing (where I define crossing as simply touching the diagonal) ?
Sep 13, 2010 at 11:25 history answered Robin Chapman CC BY-SA 2.5