On percolation in one-dimensional stable Poisson graphs. (English) Zbl 1358.60009

Björklund, Johan; Falgas-Ravry, Victor; Holmgren, Cecilia

Electron. Commun. Probab. 20, Paper No. 50, 6 p. (2015).

This paper studies percolation in one-dimensional stable Poisson graphs. Equip each point \(x\) of a homogeneous Poisson point process on the real line with \(D_x\) edge stubs, where \(D_x\) are i.i.d. positive integer-valued random variables with distribution \(\mu\). The one-dimensional stable Poisson graph \(G_1(\mu)\) can be obtained by applying the stable multi-matching scheme mutatis mutandis. It is shown that if \(\mu(\{n\in\mathbb{N}:n\geq20\cdot3^i\})\geq2^{-i}\) for all but finitely many \(i\), then a.s. the graph \(G_1(\mu)\) contains an infinite path. Percolation may occur a.s. even if \(\mu\) has support over odd integers. Moreover, for any \(\varepsilon>0\) there is a distribution \(\mu\) satisfying \(\mu(\{1\})>1-\varepsilon\), but percolation a.s. occurs.

Reviewer: Yilun Shang (Shanghai)

MSC:

60C05	Combinatorial probability
60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
05C80	Random graphs (graph-theoretic aspects)

Keywords:

Poisson process; random graph; matching; percolation