Growth and isoperimetric profile of planar graphs
Authors:
Itai Benjamini and Panos Papasoglu
Journal:
Proc. Amer. Math. Soc. 139 (2011), 41054111
MSC (2010):
Primary 53C20, 53C23, 05C10
Published electronically:
March 17, 2011
MathSciNet review:
2823055
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a planar graph such that the volume function of satisfies for some constant . Then for every vertex of and , there is a domain such that , and .
 1.
Jan
Ambjørn, Bergfinnur
Durhuus, and Thordur
Jonsson, Quantum geometry, Cambridge Monographs on
Mathematical Physics, Cambridge University Press, Cambridge, 1997. A
statistical field theory approach. MR 1465433
(98i:82001)
 2.
O.
Angel, Growth and percolation on the uniform infinite planar
triangulation, Geom. Funct. Anal. 13 (2003),
no. 5, 935–974. MR 2024412
(2005b:60015), http://dx.doi.org/10.1007/s0003900304365
 3.
Omer
Angel and Oded
Schramm, Uniform infinite planar triangulations, Comm. Math.
Phys. 241 (2003), no. 23, 191–213. MR 2013797
(2005b:60021), http://dx.doi.org/10.1007/9781441996756_16
 4.
I. Benjamini and N. Curien, On limits of graphs sphere packed in Euclidean space and applications, European J. Combinatorics, to appear (2010). http://arxiv.org/abs/0907.2609
 5.
Itai
Benjamini and Oded
Schramm, Recurrence of distributional limits of finite planar
graphs, Electron. J. Probab. 6 (2001), no. 23, 13 pp.
(electronic). MR
1873300 (2002m:82025), http://dx.doi.org/10.1214/EJP.v696
 6.
B.
H. Bowditch, A short proof that a subquadratic isoperimetric
inequality implies a linear one, Michigan Math. J. 42
(1995), no. 1, 103–107. MR 1322192
(96b:20046), http://dx.doi.org/10.1307/mmj/1029005156
 7.
N. Curien, L. Menard and G. Miermont, The uniform infinite planar quadrangulation seen from infinity. In preparation.
 8.
William
Fulton, Algebraic topology, Graduate Texts in Mathematics,
vol. 153, SpringerVerlag, New York, 1995. A first course. MR 1343250
(97b:55001)
 9.
Alexander
Grigor’yan, Heat kernel and analysis on manifolds,
AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical
Society, Providence, RI, 2009. MR 2569498
(2011e:58041)
 10.
M. Krikun, Uniform infinite planar triangulation and related timereversed critical branching process, arXiv:math/0311127
 11.
M. Krikun, Local structure of random quadrangulations, arXiv:math/0512304
 12.
Panos
Papasoglu, Cheeger constants of surfaces and
isoperimetric inequalities, Trans. Amer. Math.
Soc. 361 (2009), no. 10, 5139–5162. MR 2515806
(2010m:20064), http://dx.doi.org/10.1090/S0002994709048156
 1.
 J. Ambjørn, B. Durhuus and T. Jonsson, Quantum geometry. A statistical field theory approach. Cambridge University Press, Cambridge (1997). MR 1465433 (98i:82001)
 2.
 O. Angel, Growth and percolation on the uniform infinite planar triangulation, Geometric and Functional Analysis 13, 935974 (2003). MR 2024412 (2005b:60015)
 3.
 O. Angel and O. Schramm, Uniform infinite planar triangulations, Comm. Math. Phys. 241, 191213 (2003). MR 2013797 (2005b:60021)
 4.
 I. Benjamini and N. Curien, On limits of graphs sphere packed in Euclidean space and applications, European J. Combinatorics, to appear (2010). http://arxiv.org/abs/0907.2609
 5.
 I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6, 13 pp. (2001). MR 1873300 (2002m:82025)
 6.
 B. Bowditch, A short proof that a subquadratic isoperimetric inequality implies a linear one, Michigan Math. J. 42 (1991) no. 1, pp. 103107. MR 1322192 (96b:20046)
 7.
 N. Curien, L. Menard and G. Miermont, The uniform infinite planar quadrangulation seen from infinity. In preparation.
 8.
 W. Fulton, Algebraic Topology, Springer, 1995. MR 1343250 (97b:55001)
 9.
 A. Grigor'yan, Heat Kernel and Analysis on Manifolds. American Mathematical Society (2009). MR 2569498
 10.
 M. Krikun, Uniform infinite planar triangulation and related timereversed critical branching process, arXiv:math/0311127
 11.
 M. Krikun, Local structure of random quadrangulations, arXiv:math/0512304
 12.
 P. Papasoglu, Cheeger constants of surfaces and isoperimetric inequalities, Trans. Amer. Math. Soc. 361 (2009), 51395162. MR 2515806
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
53C20,
53C23,
05C10
Retrieve articles in all journals
with MSC (2010):
53C20,
53C23,
05C10
Additional Information
Itai Benjamini
Affiliation:
Department of Mathematics, Weizmann Institute, Rehovot, 76100, Israel
Email:
itai.benjamini@weizmann.ac.il
Panos Papasoglu
Affiliation:
Mathematical Institute, University of Oxford, 2429 St Giles’, Oxford, OX1 3LB, United Kingdom
Email:
papazoglou@maths.ox.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299392011108104
PII:
S 00029939(2011)108104
Received by editor(s):
April 24, 2010
Received by editor(s) in revised form:
September 24, 2010
Published electronically:
March 17, 2011
Communicated by:
Jianguo Cao
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
