Paper detail

Return to the Poissonian City

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial / final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination / source. Suppose further that connections are established using "near-geodesics", constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near-geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In earlier work ("Geodesics and flows in a Poissonian city", Annals of Applied Probability, 21(3), 801--842, 2011) it was shown that a scaled version of this flow had asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two "seminal curves" defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to zero with increasing computational effort.