Title: Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension,
Abstract: We consider the one-dimensional model of a spin glass with independent
Gaussian-distributed random interactions, that have mean zero and variance
$1/|i-j|^{2\sigma}$, between the spins at sites $i$ and $j$ for all $i\neq j$.
It is known that, for $\sigma>1$, there is no phase transition at any non-zero
temperature in this model. We prove rigorously that, for $\sigma>3/2$, any
Newman-Stein metastate for the ground states (i.e.\ the frequencies with which
distinct ground states are observed in finite size samples in the limit of
infinite size, for given disorder) is trivial and unique. In other words, for
given disorder and asymptotically at large sizes, the same ground state, or its
global spin flip, is obtained (almost) always. The proof consists of two parts:
one is a theorem (based on one by Newman and Stein for short-range
two-dimensional models), valid for all $\sigma>1$, that establishes triviality
under a convergence hypothesis on something similar to the energies of domain
walls, and the other (based on older results for the one-dimensional model)
establishes that the hypothesis is true for $\sigma>3/2$. In addition, we
derive heuristic scaling arguments and rigorous exponent inequalities which
tend to support the validity of the hypothesis under broader conditions. The
constructions of various metastates are extended to all values $\sigma>1/2$.
Triviality of the metastate in bond-diluted power-law models for $\sigma>1$ is
proved directly.