Abstract
We obtain the Brownian net of [24] as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous gene through such a population.
Highlights
The Brownian net, introduced in [24], arises as the scaling limit of a system of branching and coalescing random walk paths
Theorem 3.5, is a contribution in this direction. It establishes an appropriate scaling under which the paths traced out by a system of branching and coalescing continuous time and space random walks in one spatial dimension converges to the Brownian net
Our primary concern in this paper is the dual process of the Spatial Λ-Fleming-Viot process with selection (SΛFVS), a system of branching and coalescing paths that encodes all the potential ancestors of individuals in a sample from the population
Summary
The Brownian net, introduced in [24], arises as the scaling limit of a system of branching and coalescing random walk paths. Theorem 3.5, is a contribution in this direction It establishes an appropriate scaling under which the paths traced out by a system of branching and coalescing continuous time and space random walks in one spatial dimension converges to the Brownian net. We believe (and [4] provides strong supporting evidence) that the random walks can even be allowed to jump over one another and the only effect on the limiting object is a simple scaling of time (given by one minus the crossing probability of ‘nearby’ paths) This would mirror the results of [20], in which systems of coalescing non-simple random walks with crossing paths are shown to converge to the Brownian web.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
