| Summary: | Packing spheres efficiently in large dimension $d$ is a particularly
difficult optimization problem. In this paper we add an isotropic interaction
potential to the pure hard-core repulsion, and show that one can tune it in
order to maximize a lower bound on packing density. Our results suggest that
exponentially many (in the number of particles) distinct disordered sphere
packings can be effectively constructed by this method, up to a packing
fraction close to $7\, d\, 2^{-d}$. The latter is determined by solving the
inverse problem of maximizing the dynamical glass transition over the space of
the interaction potentials. Our method crucially exploits a recent exact
formulation of the thermodynamics and the dynamics of simple liquids in
infinite dimension.
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