On the Spectral Expansion of Monotone Subsets of the Hypercube
Bibliographic record
Abstract
We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset $A\subseteq\{0,1\}^{n}$ of density $μ(A)$, the previous best lower bound on the spectral gap, due to Cohen, was $γ\gtrsim μ(A)/n^{2}$, improving upon the earlier bound $γ\gtrsim μ(A)^{2}/n^{2}$ established by Ding and Mossel. In this paper, we prove the optimal lower bound $γ\gtrsim μ(A)/n$. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from $O(n^{3})$, as shown by Ding and Mossel, to $O(n^{2})$. Along the way, we develop two new inequalities that may be of independent interest: (1)~a directed $L^{2}$-Poincaré inequality on the hypercube, and (2)~an ``approximate'' FKG inequality for monotone sets.
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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) | 0.000 | 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".