Abstract
We study scale-invariant Rayleigh Random Flights (“RRF”) in random environments given by planar Scale-Invariant Random Spatial Networks (“SIRSN”) based on speed-marked Poisson line processes. A natural one-parameter family of such RRF (with scale-invariant dynamics) can be viewed as producing “randomly-broken local geodesics” on the SIRSN; we aim to shed some light on a conjecture that a (non-broken) geodesic on such a SIRSN will never come to a complete stop en route. (If true, then all such geodesics can be represented as doubly-infinite sequences of sequentially connected line segments. This would justify a natural procedure for computing geodesics.) The family of these RRF (“SIRSN-RRF”), is introduced via a novel axiomatic theory of abstract scattering representations for Markov chains (itself of independent interest). Palm conditioning (specifically the Mecke-Slivnyak theorem for Palm probabilities of Poisson point processes) and ideas from the ergodic theory of random walks in random environments are used to show that at a critical value of the parameter the speed of the scale-invariant SIRSN-RRF neither diverges to infinity nor tends to zero, thus supporting the conjecture.
Highlights
Aldous and Ganesan (2013) and Aldous (2014) introduced the notion of ScaleInvariant Random Spatial Networks (SIRSN), motivated by the ubiquitous navigational tool of online maps (Google Maps, Bing Maps, OpenStreetMap)
It is shown that the relative environment viewed from the SIRSN-Rayleigh Random Flights (RRF) is ergodic stationary, and that there exists a critical SIRSNRRF whose speed process is neighborhood-recurrent
We offer this as evidence in favour of Conjecture 1.6, that Π-geodesics in a Poisson line SIRSN never come to a complete halt, and can be constructed using doubly infinite sequences of segments taken from the Poisson line SIRSN
Summary
Aldous and Ganesan (2013) and Aldous (2014) introduced the notion of ScaleInvariant Random Spatial Networks (SIRSN), motivated by the ubiquitous navigational tool of online maps (Google Maps, Bing Maps, OpenStreetMap). A SIRSN is a random mechanism that generates networks built out of almost surely unique random routes between specified locations, required both to deliver scale-invariant statistics and to ensure considerable route-sharing between different routes. It is easy to produce random networks with translation- and isotropyinvariant statistics: the challenge is to find route-finding models which are statistically invariant under change of scale. This can be expressed as the conjecture that geodesics on such a SIRSN will never come to a complete stop en route If this conjecture is true, it justifies the natural approximation of geodesics using finite-line approximations to the SIRSN. Motivated by these considerations, this paper characterizes and describes a natural one-parameter family of random flight processes on the SIRSN. In addition the following axioms must be satisfied: 1.1.1 Similarity-invariant statistics: For each Euclidean similarity S (translation, rotation and scaling dilation), the networks N (Sx1, . . . , Sxn) and S N (x1, . . . , xn) have the same statistical law
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