For non-nestling and marginally nestling random walks in i.i.d. random environments, the quenched large deviation rate of returning to the origin is realized by traveling to a rare ball whose environment approximates an optimal periodic environment.
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Large deviations at the origin of random walk in random environment
For non-nestling and marginally nestling random walks in i.i.d. random environments, the quenched large deviation rate of returning to the origin is realized by traveling to a rare ball whose environment approximates an optimal periodic environment.