Abstract
By generalizing the path method, we show that nonlinear spectral gaps of a finite connected graph are uniformly bounded from below by a positive constant which is independent of the target metric space. We apply our result to an \(r\)-ball \(T_{d,r}\) in the \(d\)-regular tree, and observe that the asymptotic behavior of nonlinear spectral gaps of \(T_{d,r}\) as \(r\rightarrow \infty \) does not depend on the target metric space, which is in contrast to the case of a sequence of expanders. We also apply our result to the \(n\)-dimensional Hamming cube \(H_n\) and obtain an estimate of its nonlinear spectral gap with respect to an arbitrary metric space, which is asymptotically sharp as \(n\rightarrow \infty \).
Similar content being viewed by others
References
Bourgain, J.: On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math. 52(1–2), 46–52 (1985)
Bourgain, J.: The metrical interpretation of superreflexivity in Banach spaces. Israel J. Math. 56, 222–230 (1986)
Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1(1), 36–61 (1991)
Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible Markov chains. Ann. Appl. Prob. 3(3), 696–730 (1993)
Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13(1), 73–146 (2003)
Gromov, M.: \({\rm CAT}(\kappa )\)-spaces: construction and concentration. J. Math. Sci. (N. Y.) 119(2), 178–200 (2004)
Fujiwara, K., Toyoda, T.: Random groups have fixed points on CAT(0) cube complexes. Proc. Am. Math. Soc. 140(2012), 1023–1031 (2012)
Izeki, H., Kondo, T., Nayatani, S.: Fixed-point property of random groups. Ann. Global Anal. Geom. 35(4), 363–379 (2009)
Izeki, H., Kondo, T., Nayatani, S.: \(N\)-step energy of maps and fixed-point property of random groups. Groups Geom. Dyn. 6(4), 701–736 (2012)
Izeki, H., Nayatani, S.: Combinatorial harmonic maps and discrete-group actions on Hadamard spaces. Geom. Dedicata 114, 147–188 (2005)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18, 1149–1178 (1989)
Kondo, T.: \({\rm CAT}(0)\) spaces and expanders. Math. Z. 271(1–2), 343–355 (2012)
Kondo, T., Toyoda, T.: Symmetry of optimal realizations with respect to nonlinear spectral gaps. In preparation
Mendel, M., Naor, A.: Towards a calculus for non-linear spectral gaps. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA, January 2010), pp. 236–255 (2010)
Mendel, M., Naor, A.: Nonlinear spectral calculus and super-expanders. Publ. Math. Inst. Hautes Études Sci. 119, 1–95 (2014)
Naor, A., Silberman, L.: Poincaré inequalities, embeddings, and wild groups. Compositio Math. 147(5), 1546–1572 (2011)
Pansu, P.: Superrigidité géométrique et applications harmoniques. Séminaires et congrès, vol. 18, pp. 373–420. Soc. Math. France, Paris (2008)
Quastel, J.: Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45(6), 623–675 (1992)
Saloff-Coste, L.: Lectures on finite Markov chains. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Mathematics, vol. 1665, pp. 301–413. Springer, Berlin (1997)
Toyoda, T.: \({\rm CAT}(0)\) spaces on which certain type of singularity is bounded. Kodai Math. J. 33, 398–415 (2010)
Toyoda, T.: Fixed point property for a geodesically complete cocompact \({\rm CAT}(0)\) space. Preprint (2010)
Wang, M.-T.: Generalized harmonic maps and representations of discrete groups. Commun. Anal. Geom. 8(3), 545–563 (2000)
Acknowledgments
The authors would like to thank Professor Hiroyasu Izeki for valuable discussions concerning Proposition 2.2. We are also grateful to the anonymous referee for their careful reading of this paper and many helpful suggestions. The first author was supported by JST, CREST, “A mathematical challenge to a new phase of material sciences”.
Author information
Authors and Affiliations
Corresponding author
Appendix: Added in proof
Appendix: Added in proof
After we have completed this work, we found a uniform lower estimate of the nonlinear spectral gap for a regular graph has been obtained by Mendel and Naor [15, Lemma 2.1]. According to their estimate we have
However, this is not sharp by Proposition 3.1. Their estimate cannot be applied for \(T_{d,r}\) since this is not a regular graph.
Rights and permissions
About this article
Cite this article
Kondo, T., Toyoda, T. Uniform Estimates of Nonlinear Spectral Gaps. Graphs and Combinatorics 31, 1517–1530 (2015). https://doi.org/10.1007/s00373-014-1457-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1457-6