Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T01:48:35.185Z Has data issue: false hasContentIssue false
  • English
  • Français

Unimodular random trees

Published online by Cambridge University Press:  20 August 2013

ITAI BENJAMINI ,
RUSSELL LYONS  and
ODED SCHRAMM
ITAI BENJAMINI
Affiliation:
Mathematics Department, The Weizmann Institute of Science, Rehovot 76100, Israel email Itai.Benjamini@weizmann.ac.il
RUSSELL LYONS
Affiliation:
Department of Mathematics, 831 E 3rd St, Indiana University, Bloomington, IN 47405-7106, USA email rdlyons@indiana.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also note that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 359 - 373
Copyright
© Cambridge University Press, 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenman, M. and Warzel, S.. The canopy graph and level statistics for random operators on trees. Math. Phys. Anal. Geom. 9 (2006), 291–333.Google Scholar
Aldous, D. J. and Lyons, R.. Processes on unimodular random networks. Electron. J. Probab. 12 (54) (2007), 1454–1508 (electronic).Google Scholar
Aldous, D. J. and Steele, J. M.. The objective method: probabilistic combinatorial optimization and local weak convergence. Probability on Discrete Structures (Encyclopaedia Mathematical Sciences, 110). Ed. Kesten, H.. Springer, Berlin, 2004, pp. 1–72.Google Scholar
Angel, O. and Schramm, O.. Uniform infinite planar triangulations. Comm. Math. Phys. 241 (2003), 191–213.Google Scholar
Barlow, M. T. and Masson, R.. Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab. 38 (2010), 2379–2417.CrossRefGoogle Scholar
Benjamini, I. and Curien, N.. Ergodic theory on stationary random graphs. Electron. J. Probab. 17 (2012), Article 93, 20 pp. (electronic).CrossRefGoogle Scholar
Benjamini, I., Lyons, R., Peres, Y. and Schramm, O.. Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1999), 29–66.CrossRefGoogle Scholar
Benjamini, I., Lyons, R. and Schramm, O.. Percolation perturbations in potential theory and random walks. Random Walks and Discrete Potential Theory (Symposia Mathematica, XXXIX). Eds. Picardello, M. and Woess, W.. Cambridge University Press, Cambridge, 1999, pp. 56–84. Papers from the workshop held in Cortona, 1997.Google Scholar
Benjamini, I. and Schramm, O.. Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 (2001), 487–507.Google Scholar
Benjamini, I. and Schramm, O.. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (23) (2001), 13 pp. (electronic).Google Scholar
Bollobás, B. and Riordan, O.. Sparse graphs: metrics and random models. Random Structures Algorithms 39 (2011), 1–38.Google Scholar
Bowen, L.. Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102 (2003), 213–236.CrossRefGoogle Scholar
Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1 (1981), 431–450 (1982).Google Scholar
Elek, G.. On the limit of large girth graph sequences. Combinatorica 30 (2010), 553–563.Google Scholar
Elek, G. and Lippner, G.. Sofic equivalence relations. J. Funct. Anal. 258 (2010), 1692–1708.Google Scholar
Gaboriau, D. and Lyons, R.. A measurable-group-theoretic solution to von Neumann’s problem. Invent. Math. 177 (2009), 533–540.CrossRefGoogle Scholar
Garland, H. and Raghunathan, M. S.. Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups. Ann. of Math. (2) 92 (1970), 279–326.CrossRefGoogle Scholar
Hjorth, G.. A lemma for cost attained. Ann. Pure Appl. Logic 143 (2006), 87–102.CrossRefGoogle Scholar
Kapovich, I. and Benakli, N.. Boundaries of hyperbolic groups. Combinatorial and Geometric Group Theory (Contemporary Mathematics, 296). Eds. Cleary, S., Gilman, R., Myasnikov, A. G. and Shpilrain, V.. American Mathematical Society, Providence, RI, 2002, pp. 39–93. Papers from the AMS Special Sessions on Combinatorial Group Theory and on Computational Group Theory held in New York, November 4–5, 2000 and in Hoboken, NJ, April 28–29, 2001.CrossRefGoogle Scholar
Lovász, L.. Large Networks and Graph Limits. American Mathematical Society, Providence, RI, 2012.Google Scholar
Lyons, R. and Peres, Y.. Probability on Trees and Networks. Cambridge University Press, 2013, in preparation. Current version available at http://mypage.iu.edu/~rdlyons/.Google Scholar
Pemantle, R.. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 (1991), 1559–1574.Google Scholar
Raghunathan, M. S.. Discrete Subgroups of Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68). Springer, New York, 1972.CrossRefGoogle Scholar
Teplyaev, A.. Spectral analysis on infinite Sierpiński gaskets. J. Funct. Anal. 159 (1998), 537–567.Google Scholar