Annealed Invariance Principle for Random Walks on Random Graphs Generated by Point Processes in $R^d$

A. Rousselle

2016, v.22, №4, 653-696

ABSTRACT

We consider simple random walks on random graphs embedded in $\mathbb{R}^d$ and generated by point processes such as Delaunay triangulations, Gabriel graphs and the creek-crossing graphs. Under suitable assumptions on the point process, we show an annealed invariance principle for these random walks. These results hold for a large variety of point processes including Poisson point processes, Mat\'ern cluster and Mat\'ern hardcore processes which have respectively clustering and repulsiveness properties. The proof relies on the use the process of the environment seen from the particle. It allows to reconstruct the original process as an additive functional of a Markovian process under the annealed measure.

Keywords: random walk in random environment; Delaunay triangulation; Gabriel graph; point process; annealed invariance principle; environment seen from the particle; electrical network

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