194,493 research outputs found
Cycle density in infinite Ramanujan graphs
We introduce a technique using nonbacktracking random walk for estimating the
spectral radius of simple random walk. This technique relates the density of
nontrivial cycles in simple random walk to that in 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 Ab\'{e}rt,
Glasner and Vir\'{a}g. We show that an analogous result holds for all regular
graphs: the frequency of times spent by 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
Ab\'{e}rt, Glasner and Vir\'{a}g, showing that on an infinite Ramanujan graph,
the probability that simple random walk encounters a short cycle tends to 0
a.s. as the time tends to infinity.Comment: Published at http://dx.doi.org/10.1214/14-AOP961 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)
Following the derivation of the trace formulae in the first paper in this
series, we establish here a connection between the spectral statistics of
random regular graphs and the predictions of Random Matrix Theory (RMT). This
follows from the known Poisson distribution of cycle counts in regular graphs,
in the limit that the cycle periods are kept constant and the number of
vertices increases indefinitely. The result is analogous to the so called
"diagonal approximation" in Quantum Chaos. We also show that by assuming that
the spectral correlations are given by RMT to all orders, we can compute the
leading deviations from the Poisson distribution for cycle counts. We provide
numerical evidence which supports this conjecture.Comment: 15 pages, 5 figure
Quantitative Small Subgraph Conditioning
We revisit the method of small subgraph conditioning, used to establish that
random regular graphs are Hamiltonian a.a.s. We refine this method using new
technical machinery for random -regular graphs on vertices that hold not
just asymptotically, but for any values of and . This lets us estimate
how quickly the probability of containing a Hamiltonian cycle converges to 1,
and it produces quantitative contiguity results between different models of
random regular graphs. These results hold with held fixed or growing to
infinity with . As additional applications, we establish the distributional
convergence of the number of Hamiltonian cycles when grows slowly to
infinity, and we prove that the number of Hamiltonian cycles can be
approximately computed from the graph's eigenvalues for almost all regular
graphs.Comment: 59 pages, 5 figures; minor changes for clarit
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