Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane

TL;DR

This paper proves that for the Edwards-Anderson Ising spin glass model on a half-plane, almost every realization of couplings results in a unique ground state pair, establishing a form of ground state uniqueness.

Contribution

It demonstrates the existence of the infinite-volume joint distribution of couplings and ground states without subsequence limits, and shows almost sure uniqueness of ground states for the model.

Findings

Existence of the joint distribution $K(J, eta)$ without subsequence limits.

Almost sure support on a single ground state pair for given couplings.

Applicability to a wide class of coupling distributions, including Gaussian.

Abstract

We consider the Edwards-Anderson Ising spin glass model on the half-plane $Z \times Z^+$ with zero external field and a wide range of choices, including mean zero Gaussian, for the common distribution of the collection J of i.i.d. nearest neighbor couplings. The infinite-volume joint distribution $K(J,\alpha)$ of couplings J and ground state pairs $\alpha$ with periodic (respectively, free) boundary conditions in the horizontal (respectively, vertical) coordinate is shown to exist without need for subsequence limits. Our main result is that for almost every J, the conditional distribution $K(\alpha|J)$ is supported on a single ground state pair.