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The boundary of a square tiling of a graph coincides with the Poisson boundary

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Abstract

Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219–1238, 1996), we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which arises from a discrete version of Riemann’s mapping theorem. This implies a conjecture of Northshield (Potential Anal 2(4):299–314, 1993). Some of our technique apply to the non-planar case and might have further applications. When the graph is also hyperbolic then, under mild conditions, we prove the equivalence of several boundary constructions.

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Notes

  1. See Sect. 2 for definitions.

  2. These two terms were coined by the author; see Sect. 8 for definitions.

  3. Recently announced by Russell Lyons and Yuval Peres.

  4. Some authors apply Gelfand-theoretic techniques on the space of bounded harmonic functions to give an abstract definition of the Poisson boundary, and one could try to do the same using the sharp harmonic functions; we will refrain from this since our aim is not to provide yet another definition of the Poisson boundary.

  5. Feller [18, § 4] defines union and intersection operations of pairs of arbitrary bounded harmonic functions, and notes that these operations endow their space with a lattice structure (thanks to Yves Derriennic for this remark); when these functions are sharp, it is easy to check that our concepts coincide.

  6. Northshield requires instead that every cycle in \(G\ \)surrounds only finitely many vertices; that this is equivalent to (i) as proved in [39, Lemma 7.1].

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Acknowledgments

I am grateful to Omer Angel, Ori Gurel-Gurevich and Asaf Nachmias for pointing out some shortcommings in an earlier draft of this paper, and to Vadim Kaimanovich and Itai Benjamini for valuable discussions. I am also grateful to Yuval Peres for important remarks, improving in particular the definition of an M-boundary.

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  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Agelos Georgakopoulos

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  1. Agelos Georgakopoulos
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Correspondence to Agelos Georgakopoulos.

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A. Georgakopoulos was supported by EPSRC Grant EP/L002787/1 and FWF Grant P-24028-N18.

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Georgakopoulos, A. The boundary of a square tiling of a graph coincides with the Poisson boundary. Invent. math. 203, 773–821 (2016). https://doi.org/10.1007/s00222-015-0601-0

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