Hitting probabilities of Gaussian random fields and collision of eigenvalues of random matrices
Bibliographic record
Abstract
Let X = { X ( t ) , t ∈ R N } X= \{X(t), t \in \mathbb {R}^N\} be a centered Gaussian random field with values in R d \mathbb {R}^d satisfying certain conditions and let F ⊂ R d F \subset \mathbb {R}^d be a Borel set. In our main theorem, we provide a sufficient condition for F F to be polar for X X , i.e. P ( X ( t ) ∈ F for some t ∈ R N ) = 0 \mathbb P\big ( X(t) \in F \text { for some } t \in \mathbb {R}^N\big ) = 0 , which improves significantly the main result in Dalang et al. [Ann. Probab. 45 (2017), pp. 4700–4751], where the case of F F being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [Random Matrices Theory Appl. 9 (2020), p. 26] and Song et al. [J. Math. Anal. Appl. 502 (2021), p. 22].
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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 | 0.004 | 0.052 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.025 | 0.003 |
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".