We prove that in any dimension $n$ there exists an origin-symmetric ellipsoid ${\mathcal{E}} \subset {\mathbb{R}}^{n}$ of volume $c n^{2} $ that contains no points of ${\mathbb{Z}}^{n}$ other than the origin, where $c > 0$ is a universal constant. Equivalently, there exists a lattice sphere packing in ${\mathbb{R}}^{n}$ whose density is at least $cn^{2} \cdot 2^{-n}$ . Previously known constructions of sphere packings in ${\mathbb{R}}^{n}$ yielded densities of at most $C n \log n \cdot 2^{-n}$ . Our proof utilizes a stochastically evolving ellipsoid that accumulates at least $c n^{2}$ lattice points on its boundary, while containing no lattice points in its interior except for the origin.
