Abstract
We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we give an explicit bound in total variation distance for the approximation. Using this result, we calculate limiting distributions of linear eigenvalue statistics for random regular graphs. Previous results on the distribution of cycle counts by McKay, Wormald, and Wysocka (2004) used the method of switchings, a combinatorial technique for asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and demonstrates an interesting connection between the two techniques.
Highlights
Suppose that λ1, . . . , λn are the eigenvalues of an n × n random matrix
The random variable n i=1 f for a given function f is known as a linear eigenvalue statistic, and it is a common object of study in random matrix theory, typically as n tends to infinity
The bare-hands switching arguments used there might be useful to anyone who needs a sharp bound on a Poisson approximation at a single point
Summary
Suppose that λ1, . . . , λn are the eigenvalues of an n × n random matrix. The random variable n i=1 f (λi) for a given function f is known as a linear eigenvalue statistic, and it is a common object of study in random matrix theory, typically as n tends to infinity. The strongest results on the cycle counts of a random regular graph came in [26], where the Poisson approximation was shown to hold even as d = d(n) and r = r(n) grow with n, so long as (d − 1)2r−1 = o(n). This is a natural boundary: in this asymptotic regime, all cycles in G of length r or less have disjoint edges, asymptotically almost surely.
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