Suppose you have the simple random walk on $L=\mathbb{Z}^n,$ and $A \subset L$ is a subset (in my case, the subset is a union of hyperplanes, so in particular infinite). There is a random variable $T(A)$ -- the expected time to get from the origin to $A.$ The question is: is there some standard technology to upper bound the expectation of $T(A)?$ Any particularly recommended things to read?
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asked Sep 3, 2012 at 16:20
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2$\begingroup$ Perhaps this paper will provide ideas: arxiv.org/abs/1206.6568 . "Lyapunov exponents of random walks in small random potential: the lower bound" by Mountford and Mourrat. They compute, under various conditions, "the first time at which the random walk [in $\mathbb{Z}^d$] crosses the hyperplane orthogonal to $\ell$ lying at distance $n$ from the origin." $\endgroup$– Joseph O'RourkeCommented Sep 3, 2012 at 16:58
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$\begingroup$ There is a book by Sydney Redner: A guide to first-passage processes, Cambridge UP, 2001. It rather discusses physical processes, and continuous random walks, often with n<=3, but may help to find bounds also for the discrete case and general n. $\endgroup$– Karl FabianCommented Sep 3, 2012 at 20:36
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$\begingroup$ @Karl: thanks, I will try to look it up... $\endgroup$– Igor RivinCommented Sep 3, 2012 at 22:32
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