Rayleigh Random Flights on the Poisson line SIRSN

TL;DR

This paper investigates scale-invariant Rayleigh Random Flights on Poisson line-based SIRSNs, providing new axiomatic frameworks and supporting the conjecture that geodesics do not come to a complete stop en route.

Contribution

It introduces a novel axiomatic theory for scattering representations of Markov chains and analyzes the behavior of scale-invariant RRFs on SIRSNs at critical parameters.

Findings

At a critical parameter, the RRF speed remains finite and non-zero.

Supports the conjecture that geodesics on SIRSNs do not stop en route.

Develops a new axiomatic approach for Markov chain scattering representations.

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 ("SIRSNRRF"), 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 SIRSNRRF neither diverges to infinity nor tends to zero, thus supporting the conjecture.