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

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.

### 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
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