## Growth and isoperimetric profile of planar graphs

HTML articles powered by AMS MathViewer

- by Itai Benjamini and Panos Papasoglu PDF
- Proc. Amer. Math. Soc.
**139**(2011), 4105-4111 Request permission

## Abstract:

Let $\Gamma$ be a planar graph such that the volume function of $\Gamma$ satisfies $V(2n)\leq CV(n)$ for some constant $C>0$. Then for every vertex $v$ of $\Gamma$ and $n\in \mathbb N$, there is a domain $\Omega$ such that $B(v,n)\subset \Omega$, $\partial \Omega \subset B(v, 6n)$ and $|\partial \Omega | \precsim n$.## References

- 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**, DOI 10.1017/CBO9780511524417 - O. Angel,
*Growth and percolation on the uniform infinite planar triangulation*, Geom. Funct. Anal.**13**(2003), no. 5, 935–974. MR**2024412**, DOI 10.1007/s00039-003-0436-5 - Omer Angel and Oded Schramm,
*Uniform infinite planar triangulations*, Comm. Math. Phys.**241**(2003), no. 2-3, 191–213. MR**2013797**, DOI 10.1007/978-1-4419-9675-6_{1}6 - 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
- Itai Benjamini and Oded Schramm,
*Recurrence of distributional limits of finite planar graphs*, Electron. J. Probab.**6**(2001), no. 23, 13. MR**1873300**, DOI 10.1214/EJP.v6-96 - 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**, DOI 10.1307/mmj/1029005156 - N. Curien, L. Menard and G. Miermont, The uniform infinite planar quadrangulation seen from infinity. In preparation.
- William Fulton,
*Algebraic topology*, Graduate Texts in Mathematics, vol. 153, Springer-Verlag, New York, 1995. A first course. MR**1343250**, DOI 10.1007/978-1-4612-4180-5 - Alexander Grigor’yan,
*Heat kernel and analysis on manifolds*, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. MR**2569498**, DOI 10.1090/amsip/047 - M. Krikun, Uniform infinite planar triangulation and related time-reversed critical branching process, arXiv:math/0311127
- M. Krikun, Local structure of random quadrangulations, arXiv:math/0512304
- Panos Papasoglu,
*Cheeger constants of surfaces and isoperimetric inequalities*, Trans. Amer. Math. Soc.**361**(2009), no. 10, 5139–5162. MR**2515806**, DOI 10.1090/S0002-9947-09-04815-6

## Additional Information

**Itai Benjamini**- Affiliation: Department of Mathematics, Weizmann Institute, Rehovot, 76100, Israel
- MR Author ID: 311800
- Email: itai.benjamini@weizmann.ac.il
**Panos Papasoglu**- Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom
- Email: papazoglou@maths.ox.ac.uk
- 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
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 4105-4111 - MSC (2010): Primary 53C20, 53C23, 05C10
- DOI: https://doi.org/10.1090/S0002-9939-2011-10810-4
- MathSciNet review: 2823055