Abstract
We study distributions of meeting times for finite symmetric Markov chains.
For Markov kernels defined on large state spaces which satisfy certain weak
inhomogeneity in return probabilities of points up to large numbers of steps,
we obtain approximation, with explicit error bounds, of the Laplace transforms
of some meeting times (without scaling) by ratios of Green functions closely
related to hitting times of points. In studying this approximation, we identify
some key matrix power series in Markov kernels weighted with solutions to a
discrete transport-like equation with explicit coefficients, which stems from
the viewpoint that meeting time distributions are equivalent to correlations of
some linear particle system. Our result applies in particular to random walks
on large random regular graphs. It gives a justification of the corresponding
practice, among other things, in Allen, Traulsen, Tarnita and Nowak (2012) on
approximating certain critical values for the emergence of cooperation when
mutation is present.
Users
Please
log in to take part in the discussion (add own reviews or comments).