Uniform estimates of nonlinear spectral gaps

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 $r\to\infty$ 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 $n\to\infty$.Comment: to appear in Graphs and Combinatoric

Fixed point property for a CAT(0) space which admits a proper cocompact group action

We prove that if a geodesically complete $\mathrm{CAT}(0)$ space $X$ admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of $X$ is less than $1$. Let $G$ be a finite connected graph, $\mu_1 (G)$ be the linear spectral gap of $G$, and $\lambda_1 (G,X)$ be the nonlinear spectral gap of $G$ with respect to such a $\mathrm{CAT}(0)$ space $X$. Then, the result implies that the ratio $\lambda_1 (G,X) / \mu_1 (G)$ is bounded from below by a positive constant which is independent of the graph $G$. It follows that any isometric action of a random group of the graph model on such $X$ has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point

Super-expanders and warped cones

For a Banach space $X$, we show that any family of graphs quasi-isometric to levels of a warped cone $\mathcal O_\Gamma Y$ is an expander with respect to $X$ if and only if the induced $\Gamma$-representation on $L^2(Y;X)$ has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.Comment: 15 pages; to appear in Ann. Inst. Fourier; exposition rewritten, main result slightly generalised to accommodate local spectral gap

Fast Scramblers, Horizons and Expander Graphs

We propose that local quantum systems defined on expander graphs provide a simple microscopic model for thermalization on quantum horizons. Such systems are automatically fast scramblers and are motivated from the membrane paradigm by a conformal transformation to the so-called optical metric.Comment: 22 pages, 2 figures. Added further discussion in section 3. Added reference