Dynamics of the spherical Sherrington-Kirkpatrick model and average case complexity for top eigenvectors
Bibliographic record
Abstract
In this Master’s thesis, we investigate the Langevin dynamics on the spherical Sherrington- \nKirkpatrick (SSK) model, a classical mean-field spin glass model. The first contribution \nof this thesis is the asymptotic limit of energy function of the SSK model, a critical property \nlinked to the model’s equilibrium state. The thermodynamic limit of energy of the \nsystem is characterized in terms of a system of integro-differential equations as the size of \nthe system goes to infinity. Then we look at the behavior of the limiting dynamics as the \ntime goes to infinity. This long time behavior of the energy has a phase transition. In the \nregime of below the critical inverse temperature, the limiting result is zero. In the regime \nof above the critical inverse temperature, the limiting result is a constant depending on \nthe temperature. \n \nThe second contribution of this thesis is that we analyze the complexity of the zerotemperature \nLangevin dynamics (a.k.a. the gradient descent algorithm) on the SSK model. \nWe establish lower and upper bound for the hitting time, defined as the first time required \nfor the output of the algorithm to achieve a small overlap with the eigenvector corresponding \nto the smallest eigenvalue of the Wigner matrix.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".