Electronic Journal of Probability

Equivalence of zero entropy and the Liouville property for stationary random graphs

Matías Carrasco Piaggio and Pablo Lessa

Full-text: Open access

Abstract

We prove that any rooted stationary random graph satisfying a growth condition and having positive entropy almost surely admits an infinite dimensional space of bounded harmonic functions. Applications to random infinite planar triangulations and Delaunay graphs are given.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 55, 24 pp.

Dates
Received: 20 October 2015
Accepted: 1 August 2016
First available in Project Euclid: 6 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1473188082

Digital Object Identifier
doi:10.1214/16-EJP4650

Mathematical Reviews number (MathSciNet)
MR3546392

Zentralblatt MATH identifier
1346.05276

Subjects
Primary: 05C80: Random graphs [See also 60B20] 28D20: Entropy and other invariants 60B05: Probability measures on topological spaces

Keywords
stationary graphs Liouville property Delaunay graphs random triangulations

Rights
Creative Commons Attribution 4.0 International License.

Citation

Carrasco Piaggio, Matías; Lessa, Pablo. Equivalence of zero entropy and the Liouville property for stationary random graphs. Electron. J. Probab. 21 (2016), paper no. 55, 24 pp. doi:10.1214/16-EJP4650. https://projecteuclid.org/euclid.ejp/1473188082


Export citation

References

  • [1] Michael Aizenman and Simone Warzel, The canopy graph and level statistics for random operators on trees, Math. Phys. Anal. Geom. 9 (2006), no. 4, 291–333 (2007).
  • [2] O. Angel, Growth and percolation on the uniform infinite planar triangulation, Geom. Funct. Anal. 13 (2003), no. 5, 935–974.
  • [3] Omer Angel and Oded Schramm, Uniform infinite planar triangulations, Comm. Math. Phys. 241 (2003), no. 2–3, 191–213.
  • [4] Itai Benjamini, Coarse geometry and randomness, Lecture Notes in Mathematics, vol. 2100, Springer, Heidelberg, 2011, Lecture notes from the 41st Probability Summer School held in Saint-Flour, 2011, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School].
  • [5] Itai Benjamini and Nicolas Curien, Ergodic theory on stationary random graphs, Electron. J. Probab. 17 (2012), no. 93, 1–20.
  • [6] I. Benjamini, E. Paquette, and J. Pfeffer, Anchored expansion, speed, and the hyperbolic poisson voronoi tessellation, ArXiv e-prints (2014).
  • [7] Itai Benjamini and Oded Schramm, Harmonic functions on planar and almost planar graphs and manifolds, via circle packings, Invent. Math. 126 (1996), no. 3, 565–587.
  • [8] David Blackwell, On transient Markov processes with a countable number of states and stationary transition probabilities, Ann. Math. Statist. 26 (1955), 654–658.
  • [9] N. Curien, Planar stochastic hyperbolic infinite triangulations, ArXiv e-prints (2014).
  • [10] Yves Derriennic, Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires, Ann. Inst. H. Poincaré Sect. B (N.S.) 12 (1976), no. 2, 111–129.
  • [11] Y. Derriennic, Entropie, théorèmes limite et marches aléatoires, Publications de l’Institut de Recherche Mathématique de Rennes. [Publications of the Rennes Institute of Mathematical Research], Université de Rennes I Institut de Recherche Mathématique de Rennes, Rennes, 1985.
  • [12] Joseph L. Doob, Classical potential theory and its probabilistic counterpart, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1984 edition.
  • [13] Peter Gerl, Random walks on graphs with a strong isoperimetric property, J. Theoret. Probab. 1 (1988), no. 2, 171–187.
  • [14] Robert M. Gray, Entropy and information theory. 2nd ed., 2nd ed. ed., New York, NY: Springer, 2011 (English).
  • [15] V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490.
  • [16] Vadim A. Kaimanovich, Measure-theoretic boundaries of Markov chains, $0$-$2$ laws and entropy, Harmonic analysis and discrete potential theory (Frascati, 1991), Plenum, New York, 1992, pp. 145–180.
  • [17] J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993, Oxford Science Publications.
  • [18] Jean-François Le Gall, Random trees and applications, Probab. Surv. 2 (2005), 245–311.
  • [19] Russel Lyons and Yuval Peres, Probability on trees and networks, http://mypage.iu.edu/~rdlyons/, 2014.
  • [20] Russell Lyons, Robin Pemantle, and Yuval Peres, Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure, Ergodic Theory Dynam. Systems 15 (1995), no. 3, 593–619.
  • [21] Terry Lyons, A simple criterion for transience of a reversible Markov chain, Ann. Probab. 11 (1983), no. 2, 393–402.
  • [22] Jesper Møller, Lectures on random Voronoĭ tessellations, Lecture Notes in Statistics, vol. 87, Springer-Verlag, New York, 1994.
  • [23] Atsuyuki Okabe, Barry Boots, and Kōkichi Sugihara, Spatial tessellations: concepts and applications of Voronoĭ diagrams, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1992, With a foreword by D. G. Kendall.
  • [24] M.S. Pinsker, Information and information stability of random variables and processes., Holden-Day Series in Time Series Analysis. San Francisco-London-Amsterdam: Holden-Day, Inc. XII. 243 p., 1964 (English).
  • [25] Gourab Ray, Geometry and percolation on half planar triangulations, Electron. J. Probab. 19 (2014), no. 47, 28.
  • [26] B. Virág, Anchored expansion and random walk, Geom. Funct. Anal. 10 (2000), no. 6, 1588–1605.