Precise asymptotics of some meeting times arising from the voter model on large random regular graphs

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)].