Abstract
We consider two independent stationary random walks on large random regular graphs of degree k≥3 with N vertices. On these graphs, the exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model’s diffusion approximations. The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to N(k−1)∕[2(k−2)].
Highlights
We consider two independent stationary random walks on large random regular graphs of degree k ≥ 3 with N vertices
The exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model’s diffusion approximations
The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to N (k − 1)/[2(k − 2)]
Summary
In the sense of convergence in distribution and convergence of all moments, M/N → (k − 1)e/[2(k − 2)] as N → ∞, where e denotes an exponential random variable with E[e] = 1 The proof of this result begins with the well-known property that the infinite k-regular tree is the limit of large random k-regular graphs [19, 6]. The exact symmetry leading to Kemeny’s constant is violated on random regular graphs, the asymmetry is mild, and an analog of the reduction to hitting times of points mentioned above still applies to the infinite tree This property suggests that the tree is a reference geometry for approximations.
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