Zero-Temperature Dynamics of ±J Spin Glasses¶and Related Models

Abstract:

We study zero-temperature, stochastic Ising models σ^t on Z ^d with (disordered) nearest-neighbor couplings independently chosen from a distribution μ on R and an initial spin configuration chosen uniformly at random. Given d, call μ type ℐ (resp., type ℱ) if, for every x in Z ^d, σ _x ^t flips infinitely (resp., only finitely) many times as t→∞ (with probability one) – or else mixed type ℳ. Models of type ℒ and ℳ exhibit a zero-temperature version of “local non-equilibration”. For d=1, all types occur and the type of any μ is easy to determine. The main result of this paper is a proof that for d=2, ±J models (where μ=αδ _J +(1-α)δ_{- J} ) are type ℳ, unlike homogeneous models (type ℐ) or continuous (finite mean) μ's (type ℳ). We also prove that all other noncontinuous disordered systems are type ℳ for any d≥ 2. The ±J proof is noteworthy in that it is much less “local” than the other (simpler) proof. Homogeneous and ±J models for d≥ 3 remain an open problem.