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Uniform Estimates of Nonlinear Spectral Gaps

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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 \(n\rightarrow \infty \).

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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”.

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Correspondence to Tetsu Toyoda.

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

$$\begin{aligned} \lambda (H_n , X)\ge \frac{1}{n4^n}. \end{aligned}$$

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.

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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

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