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

Histogram of intensities of features detected in positive ion mode.

<p>(A) The green histogram represents all features, in red are features with one or more associated MS/MS spectra and the white features having a FCP > 0.95 in LipidFrag. (B) Histogram of MS/MS spectra per feature in positive ion mode across all 5 technical replicates. (C) and (D) show the same for negative ion mode.</p

Overview on lipid MS1 features detected in <i>C</i>. <i>elegans</i> samples in the two respective ion modes with reliable LipidFrag results.

<p>Overview on lipid MS1 features detected in <i>C</i>. <i>elegans</i> samples in the two respective ion modes with reliable LipidFrag results.</p

Comparison of LipidFrag with LipidBlast results.

<p>Comparison of LipidFrag with LipidBlast results.</p

LipidFrag workflow and related lipid sub-classes.

<p>(A) Schematic drawing of LipidFrag workflow. MS/MS spectra from known lipids derived from lipid standard materials and from unknown lipids are subjected to MetFrag in silico fragmentation, whereby all possible precursor structures are taken into consideration. During training phase true positive identity and decoy candidates are used to calculate a 2-class classifier by which reliable results from unknown lipids can be identified. (B) Structures of detected phospholipid classes, phosphatidylethanolamines (PE, LMGP0201), phosphatidylcholines (PC, LMGP0101), phosphatidylglycerols (PG, LMGP0401), phosphatidylserines (PS, LMGP0301) and phosphatidylinositols (PI, LMGP0601) (C) Structure of triacylglycerols (TG, LMGL0301) (D) Structure of ceramides (Cer, LMSP0201) and dihydroceramides (Cer, LMSP0202).</p

Example of co-elution and overlapping of different TG species in <i>C</i>. <i>elegans</i>.

<p>Analysis of spectra derived from TGs is complicated in real samples due to overlap of several isomeric and isobaric species. The upper panel shows the MS/MS spectrum of TG(18:1/18:1/16:0) standard and the lower of the same chromatographic peak in a <i>C</i>. <i>elegans</i> lipid extract.</p

Results of the MetFrag identification and the classifier testing.

<p>Results of the MetFrag identification and the classifier testing.</p

Representation of the systematic workflow of MassTRIX for annotation of MS and transcriptomic data.

<p>Both data types are combined and submitted to KEGG via the KEGG API to obtain colored pathway maps. The coloring for transcriptomic data can be defined by the user.</p

Possible comparisons between jobs in MassTRIX.

<p>(A) Screen shot from the “Compare jobs” functionality and the obtained result page comparing jobs on pathway level. (B) Result from comparing 3 different jobs on pathway level represented in a barplot. X-Axis represents the different pathway maps and Y-Axis the number of annotated compounds on this pathway. (C) Result from comparing 3 different jobs on compound level. The file can directlybe downloaded and opened in MS Excel.</p

Example of a LipidFrag identification in <i>C</i>. <i>elegans</i> data.

<p>(A) MS/MS spectrum of m/z 764.5045 at 13.1 minutes detected in <i>C</i>. <i>elegans</i> with fragment structures annotated. (B) Close up of lower mass region (m/z 100–250). (C) Structures of the best three candidates obtained from MetFrag with result filtering using foreground class probabilities. Name, formula, MetFrag score and probability are indicated below each structure.</p