Electronic Journal of Probability

Recurrence of Distributional Limits of Finite Planar Graphs

Itai Benjamini and Oded Schramm

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Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.

Article information

Electron. J. Probab., Volume 6 (2001), paper no. 23, 13 pp.

Accepted: 19 September 2001
First available in Project Euclid: 19 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Random triangulations random walks mass trasport circle packing volume growth

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Benjamini, Itai; Schramm, Oded. Recurrence of Distributional Limits of Finite Planar Graphs. Electron. J. Probab. 6 (2001), paper no. 23, 13 pp. doi:10.1214/EJP.v6-96. https://projecteuclid.org/euclid.ejp/1461097653

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  • Jan Ambjørn, Konstantinos N. Anagnostopoulos, Lars Jensen, Takashi Ichihara, and Yoshiyuki Watabiki. Quantum geometry and diffusion. J. High Energy Phys. 11 Paper 22, 16 pp. (electronic), 1998.
  • Jan Ambjørn, Bergfinnur Durhuus, and Thordur Jonsson. Quantum geometry. Cambridge University Press, Cambridge, 1997. A statistical field theory approach.
  • Jan Ambjørn, Jakob L. Nielsen, Juri Rolf, Dimitrij Boulatov, and Yoshiyuki Watabiki. The spectral dimension of 2D quantum gravity. J. High Energy Phys. 2 Paper 10, 8 pp. (electronic), 1998.
  • L. Babai. The growth rate of vertex-transitive planar graphs. Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997), pages 564–573, New York, 1997. ACM.
  • Marcel Berger. Les placements de cercles. Pour la Science (French Scientific American), 176, 1992.
  • I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1): 29–66, 1999.
  • I. Benjamini and O. Schramm. Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126 565–587, 1996.
  • Yves Colin de Verdière. Une principe variationnel pour les empilements de cercles. Inventiones Mathematicae, 104 655–669, 1991.
  • Zhicheng Gao and Nicholas C. Wormald. The distribution of the maximum vertex degree in random planar maps. J. Combin. Theory Ser. A 89 (2): 201–230, 2000.
  • Olle Häggström. Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 (3): 1423–1436, 1997.
  • Zheng-Xu He and O. Schramm. Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 (2): 123–149, 1995.
  • Johan Jonasson and Oded Schramm. On the cover time of planar graphs. Electron. Comm. Probab. 5 85–90, 2000. paper no. 10.
  • Koebe. Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 141–164, 1936.
  • V. A. Malyshev. Probability related to quantum gravitation: planar gravitation. Uspekhi Mat. Nauk 54 (4(328)): 3–46, 1999.
  • Gareth McCaughan. A recurrence/transience result for circle packings. Proc. Amer. Math. Soc. 126 (12): 3647–3656, 1998.
  • S. Malitz and A. Papakostas. On the angular resolution of planar graphs. SIAM J. Discrete Math. 7 172–183, 1994.
  • Gary L. Miller and William Thurston. Separators in two and three dimensions. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pages 300–309. ACM, Baltimore, May 1990.
  • Burt Rodin and Dennis Sullivan. The convergence of circle packings to the Riemann mapping. J. Differential Geom. 26 (2): 349–360, 1987.
  • Horst Sachs. Coin graphs, polyhedra, and conformal mapping. Discrete Math. 134 133–138, 1994.
  • Gilles Schaeffer. Random sampling of large planar maps and convex polyhedra. In Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), pages 760–769 (electronic). ACM, New York, 1999.
  • Kenneth Stephenson. Approximation of conformal structures via circle packing. In N. Papamichael, St. Ruscheweyh, and E. B. Saff, editors, Computational Methods and Function Theory 1997, Proceedings of the Third CMFT Conference 11, pages 551–582. World Scientific, 1999.
  • Robin Thomas. Recent excluded minor theorems for graphs. In Surveys in combinatorics, 1999 (Canterbury), pages 201–222. Cambridge Univ. Press, Cambridge, 1999.
  • W. T. Tutte. A census of planar triangulations. Canad. J. Math. 14 21–38, 1962.