Expanders with respect to Hadamard spaces and random graphs

Abstract: It is shown that there exists a sequence of 3-regular graphs ${G_n}{n=1}^\infty$ and a Hadamard space $X$ such that ${G_n}{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of \cite{NS11}. ${G_n}_{n=1}^\infty$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.