  The infinite-parent spatial Lambda-Fleming-Viot (SLFV) process is a model of
random growth, in which a set evolves by the addition of balls according to
points of an underlying Poisson point process, and which was recently
introduced to study genetic diversity in spatially expanding populations. In
this article, we give asymptotics for the location and depth of the moving
interface, and identify the exact asymptotic scale of the transverse
fluctuations of geodesics. Our proofs are based on a new representation of the
infinite-parent SLFV in terms of chains of reproduction events, and on the
study of the properties of a typical geodesic. Moreover, we show that our
representation coincides with the alternative definitions of the process
considered in the literature, subject to a simple condition on the initial
state. Our results represent a novel development in the study of stochastic
growth models, and also have consequences for the study of genetic diversity in
expanding populations.
