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Cycle density in infinite Ramanujan graphs. (English) Zbl 1346.60061
Authors’ abstract: We introduce a technique using nonbacktracking random walks for estimating the spectral radius of simple random walks. This technique relates the density of nontrivial cycles in a simple random walk to that in a nonbacktracking random walk. We apply this to infinite Ramanujan graphs, which are regular graphs whose spectral radius equals that of the tree of the same degree. Kesten showed that the only infinite Ramanujan graphs that are Cayley graphs are trees. This result was extended to unimodular random rooted regular graphs by M. Abért et al., [Ann. Probab. 43, No. 6, 3337–3358 (2015; Zbl 1339.05365)]. We show that an analogous result holds for all regular graphs: the frequency of times spent by a simple random walk in a nontrivial cycle is a.s. 0 on every infinite Ramanujan graph. We also give quantitative versions of that result, which we apply to answer another question of Abért et al. [loc. cit], showing that on an infinite Ramanujan graph, the probability that a simple random walk encounters a short cycle tends to 0 a.s. as the time tends to infinity.

##### MSC:
 60G50 Sums of independent random variables; random walks 60F15 Strong limit theorems 05C81 Random walks on graphs 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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