Expanders with respect to Hadamard spaces and random graphs

It is shown that there exist a sequence of 3-regular graphs {G_n}^∞_n=1 and a Hadamard space X such that {G_n}^∞_n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of the second author and Silberman. The graphs {G_n}^∞_n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublineartime 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.