Ordering of the Tracy-Widom beta distributions and fractal dimension of the level sets of the directed landscape in the temporal direction
Why this work is in the frame
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Bibliographic record
Abstract
The first part of the thesis is related to the Tracy-Widom distribution. We give a stochastic comparison and ordering of the Tracy-Widom distribution with parameter β. In particular, we show that as β grows, the Tracy-Widom random variables get smaller modulo a multiplicative coefficient.The second part of the thesis is related to the directed landscape. The directed landscape, L, is a random ’metric’ on R2 that arises as the rescaled limit of last passage percolation. We show that the level sets of last passage percolation converge to the level sets of the directed landscape in the Euclidean Hausdorff metric. We also describe the fractal nature of the level sets of the directed landscape. In particular, we prove that the level sets of L(0,0;0,t) have Hausdorff dimension of 2/3 with positive probability. We prove this by finding matching upper and lower bounds. We provide an upper bound for the Hausdorff dimension in the usual way: by counting the number of squares that cover the level set. In the case of the lower bound, we provide sufficient conditions on the one and two-point density of any stochastic process to obtain a lower bound of the Hausdorff dimension of its level sets. This theorem generalizes for stochastic processes whose densities are not proved to exist. In that case, the conditions are on the one and two-point probability of being ε close to the level set. Then, we prove that the directed landscape satisfies the conditions on the two-point probability mentioned above. We conclude that 2/3 is also the lower bound of the level set of L(0, 0; 0, t) with positive probability.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it