Ergodic theory on stationary random graphs

被引:50
作者
Benjamini, Itai [1 ]
Curien, Nicolas [2 ]
机构
[1] Weizmann Inst Sci, IL-76100 Rehovot, Israel
[2] Dma Ens, F-75005 Paris, France
关键词
Stationary random graph; Simple random walk; Ergodic Theory; Entropy; Liouville Property; LONG-RANGE PERCOLATION; RANDOM-WALKS; INVARIANT PERCOLATION; HARMONIC-FUNCTIONS; TREES; GROWTH; PLANAR; QUADRANGULATION; HOMOGENIZATION; DIMENSION;
D O I
10.1214/EJP.v17-2401
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic functions. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.
引用
收藏
页码:1 / 20
页数:20
相关论文
共 38 条
[1]  
Cuesta FA, 2011, MONATSH MATH, V163, P389, DOI 10.1007/s00605-010-0230-z
[2]   Processes on unimodular random networks [J].
Aldous, David ;
Lyons, Russell .
ELECTRONIC JOURNAL OF PROBABILITY, 2007, 12 :1454-1508
[4]   Uniform infinite planar triangulations [J].
Angell, O ;
Schramm, O .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2003, 241 (2-3) :191-213
[5]  
AVEZ A, 1974, CR ACAD SCI A MATH, V279, P25
[6]   Group-invariant percolation on graphs [J].
Benjamini, I ;
Lyons, R ;
Peres, Y ;
Schramm, O .
GEOMETRIC AND FUNCTIONAL ANALYSIS, 1999, 9 (01) :29-66
[7]   Harmonic functions on planar and almost planar graphs and manifolds, via circle packings [J].
Benjamini, I ;
Schramm, O .
INVENTIONES MATHEMATICAE, 1996, 126 (03) :565-587
[8]   Recurrence of the Zd-valued infinite snake via unimodularity [J].
Benjamini, Itai ;
Curien, Nicolas .
ELECTRONIC COMMUNICATIONS IN PROBABILITY, 2012, 17 :1-10
[9]   On limits of graphs sphere packed in Euclidean space and applications [J].
Benjamini, Itai ;
Curien, Nicolas .
EUROPEAN JOURNAL OF COMBINATORICS, 2011, 32 (07) :975-984
[10]  
Benjamini Itai, 2001, Electron. J. Probab., V6