Ancestral circulation beyond two Fleming-Viot particles

Source abstract

Reversible motion can produce a directed circulation of surviving ancestry in a \FV\ particle system. We prove that two particles are the exact threshold for universal ancestral balance in finite systems with uniform copying. For two particles, the stationary current of surviving labels vanishes for every finite irreducible reversible Markov chain and every positive killing vector. On a fixed complete three-state graph, nonzero circulation occurs for each population size at least three on an open dense set of killing vectors of full Lebesgue measure. A single killing vector produces circulation simultaneously for every population size at least three, and such vectors form a dense $G_δ$ set of full measure. The current is divergence free and vanishes on bridges, so three sites and three particles are the smallest possible setting for circulation. We also identify the probabilistic effect of this circulation: for sufficiently rare transplants, migration followed by fixation is more likely in one direction than in the reverse direction.