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Hyperbolic and Parabolic Unimodular Random Maps

Abstract

We show that for infinite planar unimodular random rooted maps. many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini–Schramm limit of finite maps.

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References

  1. M. Aizenman, H. Kesten, and C. M. Newman. Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys., (4)111 (1987), 505–531

  2. D. Aldous and R. Lyons. Processes on unimodular random networks. Electron. J. Probab., (12)54 (2007), 1454–1508

  3. A. D. Aleksandrov and V. A. Zalgaller. Intrinsic geometry of surfaces. Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 15. American Mathematical Society, Providence, R.I. (1967)

  4. O. Angel, T. Hutchcroft, A. Nachmias, and G. Ray. Unimodular hyperbolic triangulations: Circle packing and random walk. Inventiones Mathematicae, to appear. arXiv:1501.04677 (2015)

  5. O. Angel and G. Ray. Classification of half planar maps. Ann. Probab., to appear.. arXiv:1303.6582 (2013)

  6. O. Angel and G. Ray. The half plane UIPT is recurrent. arXiv preprint arXiv:1601.00410 (2016)

  7. O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys., (2-3)241 (2003), 191–213

  8. A. F. Beardon and K. Stephenson. Circle packings in different geometries. Tohoku Mathematical Journal, Second Series, (1)43 (1991), 27–36

  9. I. Benjamini and N. Curien. Ergodic theory on stationary random graphs. Electron. J. Probab., (93)17 (2012), 20

  10. I. Benjamini, N. Curien, and A. Georgakopoulos. The Liouville and the intersection properties are equivalent for planar graphs. Electron. Commun. Probab., (42)17 (2012), 5

  11. I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab., (3)27 (1999), 1347–1356

  12. Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Group-invariant percolation on graphs. Geom. Funct. Anal. 9(1), 29–66 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  13. Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29(1), 1–65 (2001)

    MathSciNet  MATH  Google Scholar 

  14. I. Benjamini, R. Lyons, and O. Schramm. Percolation perturbations in potential theory and random walks. In Random walks and discrete potential theory (Cortona, 1997), Sympos. Math., XXXIX, pages 56–84. Cambridge Univ. Press, Cambridge (1999)

  15. I. Benjamini, R. Lyons, and O. Schramm. Unimodular random trees. arXiv:1207.1752 (2012)

  16. I. Benjamini, E. Paquette, and J. Pfeffer. Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation. arXiv:1409.4312

  17. I. Benjamini and O. Schramm. Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math., (3)126 (1996), 565–587

  18. I. Benjamini and O. Schramm. Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab., (3)24 (1996), 1219–1238,

  19. I. Benjamini and O. Schramm. Percolation in the hyperbolic plane, (2000)

  20. I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., (23)6 (2001), 1–13

  21. I. Benjamini and O. Schramm. Percolation beyond \(\mathbb{Z}^d\), many questions and a few answers [mr1423907]. In: Selected works of Oded Schramm. Volume 1, 2, Sel. Works Probab. Stat., pages 679–690. Springer, New York (2011)

  22. I. Biringer and J. Raimbault. The topology of invariant random surfaces. arxiv:1411.0561

  23. Bowen, L.: Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102, 213–236 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  24. L. Brouwer. On the structure of perfect sets of points. In: KNAW, Proceedings, volume 12, pages 1909–1910

  25. R. M. Burton and M. Keane. Density and uniqueness in percolation. Communications in mathematical physics, (3)121 (1989), 501–505

  26. B. Chen. The gauss-bonnet formula of polytopal manifolds and the characterization of embedded graphs with nonnegative curvature. Proceedings of the American Mathematical Society, (5)137 (2009), 1601–1611

  27. N. Curien. A glimpse of the conformal structure of random planar maps. arXiv:1308.1807 (2013)

  28. N. Curien. Planar stochastic hyperbolic infinite triangulations. arXiv:1401.3297 (2014)

  29. N. Curien, L. Ménard, and G. Miermont. A view from infinity of the uniform infinite planar quadrangulation. ALEA Lat. Am. J. Probab. Math. Stat., (1)10 (2013), 45–88

  30. J. Ding, J. R. Lee, and Y. Peres. Markov type and threshold embeddings. Geometric and Functional Analysis, (4)23 (2013), 1207–1229

  31. Duffin, R.J.: The extremal length of a network. J. Math. Anal. Appl. 5, 200–215 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  32. G. Elek. On the limit of large girth graph sequences. Combinatorica, (5)30 (2010), 553–563

  33. G. Elek and G. Lippner. Sofic equivalence relations. J. Funct. Anal., (5)258 (2010), 1692–1708

  34. A. Gandolfi, M. Keane, and C. Newman. Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probability Theory and Related Fields, (4)92 (1992), 511–527

  35. C. Garban. Quantum gravity and the KPZ formula [after Duplantier-Sheffield]. Astérisque, (352):Exp. No. 1052, ix, 315–354, 2013. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058

  36. J. R. Gilbert, J. P. Hutchinson, and R. E. Tarjan. A separator theorem for graphs of bounded genus. Journal of Algorithms, (3)5 (1984), 391–407

  37. J. T. Gill and S. Rohde. On the Riemann surface type of random planar maps. Rev. Mat. Iberoam., (3)29 (2013), 1071–1090

  38. O. Gurel-Gurevich and A. Nachmias. Recurrence of planar graph limits. Ann. of Math. (2), (2)177 (2013), 761–781

  39. O. Häggström. Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab., (3)25 (1997), 1423–1436

  40. O. Häggström and Y. Peres. Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab. Theory Related Fields, (2)113 (1999), 273–285

  41. Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  42. Z.-X. He and O. Schramm. Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2), (2)137 (1993), 369–406

  43. Z.-X. He and O. Schramm. Hyperbolic and parabolic packings. Discrete Comput. Geom., (2)14 (1995), 123–149

  44. Y. Higuchi. Combinatorial curvature for planar graphs. Journal of Graph Theory, (4)38 (2001), 220–229

  45. T. Hutchcroft. Interlacements and the wired uniform spanning forest. arXiv:1512.08509 (2015)

  46. T. Hutchcroft. Wired cycle-breaking dynamics for uniform spanning forests. arXiv:1504.03928 (2015)

  47. T. Hutchcroft and A. Nachmias. Uniform spanning forests of planar graphs. arxiv:1603.07320

  48. Kaimanovich, V.: Boundary and entropy of random walks in random environment. Prob. Theory and Math. Stat 1, 573–579 (1990)

    MathSciNet  Google Scholar 

  49. V. A. Kaimanovich. Hausdorff dimension of the harmonic measure on trees. Ergodic Theory and Dynamical Systems, (03)18 (1998), 631–660

  50. V. A. Kaimanovich. Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization. In: Fractals in Graz 2001, pages 145–183. Springer (2003)

  51. V. A. Kaimanovich, Y. Kifer, and B.-Z. Rubshtein. Boundaries and harmonic functions for random walks with random transition probabilities. Journal of Theoretical Probability, (3)17 (2004), 605–646

  52. V. A. Kaimanovich and A. M. Vershik. Random walks on discrete groups: boundary and entropy. The annals of robability, pages 457–490, (1983)

  53. V. A. Kaimanovich and W. Woess. Boundary and entropy of space homogeneous markov chains. Annals of probability, pages 323–363, (2002)

  54. M. Krikun. Local structure of random quadrangulations. arXiv:math/0512304 (2005)

  55. Lando, S.K., Zvonkin, A.K., Graphs on surfaces and their applications, volume 141 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, : With an appendix by Don B. Zagier, Low-Dimensional Topology, II (2004)

  56. G. F. Lawler. A self-avoiding random walk. Duke Math. J., (3)47 (1980) 655–693

  57. R. J. Lipton and R. E. Tarjan. Applications of a planar separator theorem. SIAM J. Comput., (3)9 (1980), 615–627

  58. R. Lyons, B. J. Morris, and O. Schramm. Ends in uniform spanning forests. Electron. J. Probab., (58)13 (2008), 1702–1725

  59. R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University Press, (2015). In preparation. Current version available at http://mypage.iu.edu/~rdlyons/

  60. R. Lyons, Y. Peres, and O. Schramm. Minimal spanning forests. Ann. Probab., (5)34 (2006), 1665–1692

  61. R. Lyons and O. Schramm. Indistinguishability of percolation clusters. Ann. Probab., (4)27 (1999), 1809–1836

  62. C. M. Newman and L. S. Schulman. Infinite clusters in percolation models. J. Statist. Phys., (3)26 (1981), 613–628

  63. R. Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., (4)19 (1991), 1559–1574

  64. I. Richards. On the classification of noncompact surfaces. Transactions of the American Mathematical Society, (2)106 (1963), 259–269

  65. K. Stephenson. Introduction to circle packing. Cambridge University Press, Cambridge, 2005. The theory of discrete analytic functions

  66. D. B. Wilson. Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pages 296–303. ACM, New York (1996)

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Correspondence to Asaf Nachmias.

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Angel, O., Hutchcroft, T., Nachmias, A. et al. Hyperbolic and Parabolic Unimodular Random Maps. Geom. Funct. Anal. 28, 879–942 (2018). https://doi.org/10.1007/s00039-018-0446-y

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