Uniform Estimates of Nonlinear Spectral Gaps

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 $$r$$r-ball $$T_{d,r}$$Td,r in the $$d$$d-regular tree, and observe that the asymptotic behavior of nonlinear spectral gaps of $$T_{d,r}$$Td,r as $$r\rightarrow \infty $$r�� 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 $$n$$n-dimensional Hamming cube $$H_n$$Hn and obtain an estimate of its nonlinear spectral gap with respect to an arbitrary metric space, which is asymptotically sharp as $$n\rightarrow \infty $$n��.