math.PR — Probability

The Parisi Formula for the Edwards–Anderson Model in the High-Dimensional Limit

Contributed by P. M. Aronow, Patrick Lopatto

We study the Edwards–Anderson spin glass on the torus $(\mathbb{Z}/L\mathbb{Z})^d$, with independent Gaussian couplings of variance $1/(2d)$ between nearest neighbors. We prove that, as $d\to\infty$, its free energy converges to that of the Sherrington–Kirkpatrick model, given by the Parisi formula, at every temperature and in every uniform external field, uniformly in the side length $L$. The same holds on the hypercube $\{0,1\}^d$, and the ground-state energies converge to the Sherrington–Kirkpatrick ground-state energy.