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.
Similar content being viewed by others
References
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
D. Aldous and R. Lyons. Processes on unimodular random networks. Electron. J. Probab., (12)54 (2007), 1454–1508
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)
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)
O. Angel and G. Ray. Classification of half planar maps. Ann. Probab., to appear.. arXiv:1303.6582 (2013)
O. Angel and G. Ray. The half plane UIPT is recurrent. arXiv preprint arXiv:1601.00410 (2016)
O. Angel and O. Schramm. Uniform infinite planar triangulations. Comm. Math. Phys., (2-3)241 (2003), 191–213
A. F. Beardon and K. Stephenson. Circle packings in different geometries. Tohoku Mathematical Journal, Second Series, (1)43 (1991), 27–36
I. Benjamini and N. Curien. Ergodic theory on stationary random graphs. Electron. J. Probab., (93)17 (2012), 20
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
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
Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Group-invariant percolation on graphs. Geom. Funct. Anal. 9(1), 29–66 (1999)
Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29(1), 1���65 (2001)
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)
I. Benjamini, R. Lyons, and O. Schramm. Unimodular random trees. arXiv:1207.1752 (2012)
I. Benjamini, E. Paquette, and J. Pfeffer. Anchored expansion, speed, and the hyperbolic Poisson Voronoi tessellation. arXiv:1409.4312
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
I. Benjamini and O. Schramm. Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab., (3)24 (1996), 1219–1238,
I. Benjamini and O. Schramm. Percolation in the hyperbolic plane, (2000)
I. Benjamini and O. Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., (23)6 (2001), 1–13
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)
I. Biringer and J. Raimbault. The topology of invariant random surfaces. arxiv:1411.0561
Bowen, L.: Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102, 213–236 (2003)
L. Brouwer. On the structure of perfect sets of points. In: KNAW, Proceedings, volume 12, pages 1909–1910
R. M. Burton and M. Keane. Density and uniqueness in percolation. Communications in mathematical physics, (3)121 (1989), 501–505
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
N. Curien. A glimpse of the conformal structure of random planar maps. arXiv:1308.1807 (2013)
N. Curien. Planar stochastic hyperbolic infinite triangulations. arXiv:1401.3297 (2014)
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
J. Ding, J. R. Lee, and Y. Peres. Markov type and threshold embeddings. Geometric and Functional Analysis, (4)23 (2013), 1207–1229
Duffin, R.J.: The extremal length of a network. J. Math. Anal. Appl. 5, 200–215 (1962)
G. Elek. On the limit of large girth graph sequences. Combinatorica, (5)30 (2010), 553–563
G. Elek and G. Lippner. Sofic equivalence relations. J. Funct. Anal., (5)258 (2010), 1692–1708
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
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
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
J. T. Gill and S. Rohde. On the Riemann surface type of random planar maps. Rev. Mat. Iberoam., (3)29 (2013), 1071–1090
O. Gurel-Gurevich and A. Nachmias. Recurrence of planar graph limits. Ann. of Math. (2), (2)177 (2013), 761–781
O. Häggström. Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab., (3)25 (1997), 1423–1436
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
Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)
Z.-X. He and O. Schramm. Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2), (2)137 (1993), 369–406
Z.-X. He and O. Schramm. Hyperbolic and parabolic packings. Discrete Comput. Geom., (2)14 (1995), 123–149
Y. Higuchi. Combinatorial curvature for planar graphs. Journal of Graph Theory, (4)38 (2001), 220–229
T. Hutchcroft. Interlacements and the wired uniform spanning forest. arXiv:1512.08509 (2015)
T. Hutchcroft. Wired cycle-breaking dynamics for uniform spanning forests. arXiv:1504.03928 (2015)
T. Hutchcroft and A. Nachmias. Uniform spanning forests of planar graphs. arxiv:1603.07320
Kaimanovich, V.: Boundary and entropy of random walks in random environment. Prob. Theory and Math. Stat 1, 573–579 (1990)
V. A. Kaimanovich. Hausdorff dimension of the harmonic measure on trees. Ergodic Theory and Dynamical Systems, (03)18 (1998), 631–660
V. A. Kaimanovich. Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization. In: Fractals in Graz 2001, pages 145–183. Springer (2003)
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
V. A. Kaimanovich and A. M. Vershik. Random walks on discrete groups: boundary and entropy. The annals of robability, pages 457–490, (1983)
V. A. Kaimanovich and W. Woess. Boundary and entropy of space homogeneous markov chains. Annals of probability, pages 323–363, (2002)
M. Krikun. Local structure of random quadrangulations. arXiv:math/0512304 (2005)
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)
G. F. Lawler. A self-avoiding random walk. Duke Math. J., (3)47 (1980) 655–693
R. J. Lipton and R. E. Tarjan. Applications of a planar separator theorem. SIAM J. Comput., (3)9 (1980), 615–627
R. Lyons, B. J. Morris, and O. Schramm. Ends in uniform spanning forests. Electron. J. Probab., (58)13 (2008), 1702–1725
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/
R. Lyons, Y. Peres, and O. Schramm. Minimal spanning forests. Ann. Probab., (5)34 (2006), 1665–1692
R. Lyons and O. Schramm. Indistinguishability of percolation clusters. Ann. Probab., (4)27 (1999), 1809–1836
C. M. Newman and L. S. Schulman. Infinite clusters in percolation models. J. Statist. Phys., (3)26 (1981), 613–628
R. Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., (4)19 (1991), 1559–1574
I. Richards. On the classification of noncompact surfaces. Transactions of the American Mathematical Society, (2)106 (1963), 259–269
K. Stephenson. Introduction to circle packing. Cambridge University Press, Cambridge, 2005. The theory of discrete analytic functions
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-018-0446-y