Free energy fluctuations of the two-spin spherical SK model at critical temperature
Bibliographic record
Abstract
We investigate the fluctuations of the free energy of the two-spin spherical Sherrington–Kirkpatrick model at critical temperature βc = 1. When β = 1, we find asymptotic Gaussian fluctuations with variance 16N2log(N), confirming in the spherical case a physics prediction for the SK model with Ising spins. We, furthermore, prove the existence of a critical window on the scale β=1+αlog(N)N−1/3. For any α∈R, we show that the fluctuations are at most order log(N)/N in the sense of tightness. If α → ∞ at any rate as N → ∞, then, properly normalized, the fluctuations converge to the Tracy–Widom1 distribution. If α → 0 at any rate as N → ∞ or α < 0 is fixed, the fluctuations are asymptotically Gaussian as in the α = 0 case. In determining the fluctuations, we apply a recent result of G. Lambert and E. Paquette [“Strong approximation of Gaussian beta-ensemble characteristic polynomials: The edge regime and the stochastic airy function,” arXiv:2009.05003 (2020)] on the behavior of the Gaussian-β-ensemble at the spectral edge.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".