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arXiv · 1011.3118

Linear Cover Time is Exponentially Unlikely

Itai Benjamini·Ori Gurel-Gurevich·Ben Morris
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Abstract

We show that the probability that a simple random walk covers a finite, bounded degree graph in linear time is exponentially small. More precisely, for every D and C, there exists a=a(D,C)>0 such that for any graph G, with n vertices and maximal degree D, the probability that a simple random walk, started anywhere in G, will visit every vertex of G in its first Cn steps is at most exp(-an). We conjecture that the same holds for a=a(C)>0 that does not depend on D, provided that the graph G is simple.

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Itai Benjamini, Ori Gurel-Gurevich, Ben Morris. 2010-11-13. Linear Cover Time is Exponentially Unlikely. https://arxiv.org/abs/1011.3118

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