Small spectral radius and percolation constants on non-amenable Cayley graphs
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- by Kate Juschenko and Tatiana Nagnibeda PDF
- Proc. Amer. Math. Soc. 143 (2015), 1449-1458 Request permission
Abstract:
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $\Gamma$, does there exist a generating set $S$ such that the Cayley graph $(\Gamma ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(\Gamma ,S)<p_u(\Gamma ,S)$? We show that this is true if $\Gamma$ contains an infinite normal subgroup $N$ such that $\Gamma /N$ is non-amenable. Moreover for any finitely generated group $G$ containing $\Gamma$ there exists a generating set $S’$ of $G$ such that $p_c(G,S’)<p_u(G,S’)$. In particular this applies to free Burnside groups $B(n,p)$ with $n \geq 2, p \geq 665$. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group.References
- S. I. Adyan, Random walks on free periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1139–1149, 1343 (Russian). MR 682486
- B. Bekka, P. de la Harpe, A. Valette, Kazhdan Property $(T)$, Cambridge University Press, 2008.
- Itai Benjamini, Asaf Nachmias, and Yuval Peres, Is the critical percolation probability local?, Probab. Theory Related Fields 149 (2011), no. 1-2, 261–269. MR 2773031, DOI 10.1007/s00440-009-0251-5
- Itai Benjamini and Oded Schramm, Percolation beyond $\mathbf Z^d$, many questions and a few answers, Electron. Comm. Probab. 1 (1996), no. 8, 71–82. MR 1423907, DOI 10.1214/ECP.v1-978
- Itai Benjamini and Oded Schramm, Percolation in the hyperbolic plane, J. Amer. Math. Soc. 14 (2001), no. 2, 487–507. MR 1815220, DOI 10.1090/S0894-0347-00-00362-3
- R. M. Burton and M. Keane, Density and uniqueness in percolation, Comm. Math. Phys. 121 (1989), no. 3, 501–505. MR 990777
- Jozef Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. MR 743744, DOI 10.1090/S0002-9947-1984-0743744-X
- Erling Følner, On groups with full Banach mean value, Math. Scand. 3 (1955), 243–254. MR 79220, DOI 10.7146/math.scand.a-10442
- D. Gaboriau, Invariant percolation and harmonic Dirichlet functions, Geom. Funct. Anal. 15 (2005), no. 5, 1004–1051. MR 2221157, DOI 10.1007/s00039-005-0539-2
- Olle Häggström and Yuval Peres, Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously, Probab. Theory Related Fields 113 (1999), no. 2, 273–285. MR 1676835, DOI 10.1007/s004400050208
- Paul Jolissaint, Rapidly decreasing functions in reduced $C^*$-algebras of groups, Trans. Amer. Math. Soc. 317 (1990), no. 1, 167–196. MR 943303, DOI 10.1090/S0002-9947-1990-0943303-2
- Harry Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354. MR 109367, DOI 10.1090/S0002-9947-1959-0109367-6
- Russell Lyons, Random walks and the growth of groups, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 11, 1361–1366 (English, with English and French summaries). MR 1338287
- Russell Lyons, Phase transitions on nonamenable graphs, J. Math. Phys. 41 (2000), no. 3, 1099–1126. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. MR 1757952, DOI 10.1063/1.533179
- Russell Lyons, Fixed price of groups and percolation, Ergodic Theory Dynam. Systems 33 (2013), no. 1, 183–185. MR 3009109, DOI 10.1017/S0143385711000927
- Bojan Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl. 103 (1988), 119–131. MR 943998, DOI 10.1016/0024-3795(88)90224-8
- Igor Pak and Tatiana Smirnova-Nagnibeda, On non-uniqueness of percolation on nonamenable Cayley graphs, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 495–500 (English, with English and French summaries). MR 1756965, DOI 10.1016/S0764-4442(00)00211-1
- V. L. Širvanjan, Imbedding of the group $B(\infty ,$ $n)$ in the group $B(2,$ $n)$, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 1, 190–208, 223 (Russian). MR 0414726
- A. Thom, A remark about the spectral radius. ArXiv:1306.1767
Additional Information
- Kate Juschenko
- Affiliation: École polytechnique fédérale de Lausanne, Route Cantonale, 1015 Lausanne, Switzerland
- MR Author ID: 780620
- Email: kate.juschenko@gmail.com
- Tatiana Nagnibeda
- Affiliation: Section de mathématiques, Université de Genève, 2-4, rue du Lièvre c.p. 64, 1211 Genève, Switzerland
- Email: tatiana.smirnova-nagnibeda@unige.ch
- Received by editor(s): August 29, 2013
- Published electronically: December 9, 2014
- Additional Notes: The authors acknowledge the support of the Swiss National Foundation for Scientific Research and of the Mittag-Leffler Insitute
- Communicated by: Ken Ono
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1449-1458
- MSC (2010): Primary 20F65, 97K50; Secondary 20F05
- DOI: https://doi.org/10.1090/S0002-9939-2014-12578-0
- MathSciNet review: 3314060