Absence of percolation in graphs based on stationary point processes with degrees bounded by two

TL;DR

This paper proves that certain stationary point process-based graphs with degree bounds do not contain infinite components, confirming a conjecture for specific k-nearest neighbor graphs in 2D Poisson processes.

Contribution

It extends previous results by showing non-percolation in a broad class of stationary point process graphs with degree at most two.

Findings

Graphs have no infinite connected component almost surely.

Confirms the non-percolation conjecture for 2-nearest neighbor graphs.

Extends non-percolation results to a wide class of point processes.

Abstract

We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollob\'as that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for $k=2$.