Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences
Bibliographic record
Abstract
Let X be a d-partite d-dimensional simplicial complex with parts T1,…, Tdand let μ be a distribution on the facets of X. Informally, we say (X, μ) is a path complex if for any ii, G ∈ Tj, K ∈ Tk, we have ${{\mathbb{P}}_\mu }[F,K\mid G] = {{\mathbb{P}}_\mu }[F\mid G]\cdot{{\mathbb{P}}_\mu }[K\mid G]$. We develop a new machinery with ${\mathcal{C}}$-Lorentzian polynomials to show that if all links of X of co-dimension 2 have spectral expansion at most 1/2, then X is a 1/2-local spectral expander. We then prove that one can derive fast-mixing results and log-concavity statements for top-link spectral expanders.We use our machinery to prove fast mixing results for sampling maximal flags of flats of distributive lattices (a.k.a. linear extensions of posets) subject to external fields, and to sample maximal flags of flats of "typical" modular lattices. We also use it to re-prove the Heron-Rota-Welsh conjecture and to prove a conjecture of Chan and Pak which gives a generalization of Stanley’s log-concavity theorem. Lastly, we use it to prove near optimal trickle-down theorems for "sparse complexes" such as constructions by Lubotzky-Samuels-Vishne, Kaufman-Oppenheim, and O’Donnell-Pratt.
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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 | 0.008 | 0.045 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.005 | 0.010 |
| Open science | 0.004 | 0.008 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.014 | 0.002 |
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".