Approximating Laplace transforms of meeting times for some symmetric Markov chains

TL;DR

This paper develops an approximation method for the Laplace transforms of meeting times in symmetric Markov chains, especially on large, weakly inhomogeneous state spaces, with applications to random regular graphs and evolutionary dynamics.

Contribution

It introduces explicit error bounds for approximating meeting time Laplace transforms using Green functions and matrix power series, connecting to particle system correlations.

Findings

Provides explicit error bounds for approximations.

Validates the approximation method on large random regular graphs.

Supports applications in evolutionary game theory and cooperation emergence.

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.