The Hopfield Model with Superlinearly Many Patterns

We study the Hopfield model where the ratio $\alpha$ of patterns to sites grows large. We prove that the free energy with inverse temperature $\beta$ and external field $B$ behaves like $\beta \sqrt{\alpha }+\gamma$, where $\gamma$ is the limiting free energy of the Sherrington-Kirkpatrick model with inverse temperature $\sqrt{2}\beta$ and external field $B$.