Approximate Ground States of Hypercube Spin Glasses are Near Corners

Abstract

We show that with probability exponentially close to $1$, all near-maximizers of any mean-field mixed $p$-spin glass Hamiltonian on the hypercube $[-1,1{]}^N$ are near a corner. This confirms a recent conjecture of Gamarnik and Jagannath. The proof is elementary and generalizes to arbitrary polytopes with $e^{o(N^2)}$ faces.