Proof of single-replica equivalence in short-range spin glasses

TL;DR

This paper proves that in short-range Ising spin glasses at equilibrium, all pure states in a Gibbs state share identical values for translation- and locally-invariant functions, revealing a form of single-replica equivalence.

Contribution

It establishes a rigorous proof of single-replica equivalence in short-range spin glasses, a key property previously conjectured but not proven.

Findings

All pure states have identical invariant functions.

The result applies to fixed bond realizations and Gibbs states.

Several significant applications to spin glass theory are discussed.

Abstract

We consider short-range Ising spin glasses in equilibrium at infinite system size, and prove that, for fixed bond realization and a given Gibbs state drawn from a suitable metastate, each translation- and locally-invariant function (for example, self-overlaps) of a single pure state in the decomposition of the Gibbs state takes the same value for all the pure states in that Gibbs state. We describe several significant applications to spin glasses.