Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension

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.Comment: 18 pages. v2: subsection on bond-diluted models added, few extra references. 19 pages. v3: published version; a few changes; 20 page

Additional file 3: of Relationships between drought, heat and air humidity responses revealed by transcriptome-metabolome co-analysis

Result lists of condition-specific gene expression and metabolomic mass levels estimated by linear modeling across the whole dataset. Tables show mean log2 gene expression values and log2metabolomic mass levels for each of the six environmental conditions: control, drought (D), heat with low relative air humidity (H_LrH), heat with high relative air humidity (H_HrH), drought stress combined with low air humidity heat stress (DH_LrH) and drought stress combined with high air humidity heat stress (DH_HrH). Values of all individual samples are available in the processed data file deposited at the ArrayExpress and MetaboLights databases, respectively (Availability of data and materials). (XLSX 2264 kb