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We introduce a modified Galton-Watson process using the framework of aninfinite system of particles labeled by $(x,t)$, where $x$ is the rank of theparticle born at time $t$. The key assumption concerning the offspring numbersof different particles is that they are independent, but their distributionsmay depend on the particle label $(x,t)$. For the associated system of coupledmonotone Markov chains, we address the issue of pathwise duality elucidated bya remarkable graphical representation, with the trajectories of the primaryMarkov chains and their duals coalescing together to form forest graphs on atwo-dimensional grid.

Rank-dependent Galton-Watson processes and their pathwise duals