  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.
