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A stationary planar random graph with singular stationary dual: dyadic lattice graphs

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Abstract

Dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane. We use them to give examples of random rooted graphs that are stationary for simple random walk, but whose duals have only a singular stationary measure. This answers a question of Curien and shows behaviour different from the unimodular case. The consequence is that planar duality does not combine well with stationary random graphs. We also study harmonic measure on dyadic lattice graphs and show its singularity.

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Acknowledgements

We are grateful to the referees for their careful readings and questions, which led to improved clarity of our paper.

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Correspondence to Russell Lyons.

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The work of R.L. is partially supported by the National Science Foundation under Grant DMS-1612363.

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Lyons, R., White, G. A stationary planar random graph with singular stationary dual: dyadic lattice graphs. Probab. Theory Relat. Fields (2019). https://doi.org/10.1007/s00440-019-00934-0

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Keywords

  • Unimodular
  • Random graphs
  • Whitney decomposition
  • Hyperbolic model
  • Baumslag–Solitar
  • Harmonic measure

Mathematics Subject Classification

  • Primary 05C81
  • 05C80
  • 60G50
  • Secondary 5C10
  • 60K37