Approximating Laplace transforms of meeting times for some symmetric Markov chains

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.