Statistical mechanics of the lattice sphere packing problem

We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit $d\to\infty$. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition, but leave open the possibility that a glass transition is manifested in higher dimensions.