We use extreme value statistics to assess the significance of two of the most dramatic structures in the local Universe: the Shapley supercluster and the Sloan Great Wall. If we assume that Shapley (volume ~ 1.2 x 10^5 (Mpc/h)^3) evolved from an overdense region in the initial Gaussian fluctuation field, with currently popular choices for the background cosmological model and the shape and amplitude sigma8 of the initial power spectrum, we estimate that the total mass of the system is within 20 percent of 1.8 x 10^16 Msun/h. Extreme value statistics show that the existence of this massive concentration is not unexpected if the initial fluctuation field was Gaussian, provided there are no other similar objects within a sphere of radius 200 Mpc/h centred on our Galaxy. However, a similar analysis of the Sloan Great Wall, a more distant (z ~ 0.08) and extended concentration of structures (volume ~ 7.2 x 10^5 (Mpc/h)^3) suggests that it is more unusual. We estimate its total mass to be within 20 percent of 1.2 x 10^17 Msun/h; even if it is the densest such object of its volume within z=0.2, its existence is difficult to reconcile with Gaussian initial conditions if sigma8 < 0.9. This tension can be alleviated if this structure is the densest within the Hubble volume. Finally, we show how extreme value statistics can be used to address the likelihood that an object like Shapley exists in the same volume which contains the Great Wall, finding, again, that Shapley is not particularly unusual. It is straightforward to incorporate other models of the initial fluctuation field into our formalism.
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