Is ‘being above the median’ a noise sensitive property?
Bibliographic record
Abstract
Assign independent weights to the edges of the square lattice, from the uniform distribution on { a , b } \{a,b\} for some 0 > a > b > ∞ 0>a>b>\infty . The weighted graph induces a random metric on Z 2 {\mathbb {Z}}^2 . Let T n T_n denote the distance between ( 0 , 0 ) (0,0) and ( n , 0 ) (n,0) in this metric. The distribution of T n T_n has a well-defined median. Itai Benjamini asked in 2011 if the sequence of Boolean functions encoding whether T n T_n exceeds its median is noise sensitive? In this paper we present the first progress on Benjamini’s problem. More precisely, we study the minimal weight along any path crossing an n × n n\times n -square horizontally and whose vertical fluctuation is smaller than n 1 / 22 n^{1/22} , and show that for this observable, ‘being above the median’ is a noise sensitive property.
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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 | 0.006 | 0.055 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 0.006 |
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".