Grid Entropy in Last Passage Percolation, a Variational Formula for Gibbs Free Energy, and Applications to a ”choose the best of D samples” Model
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Bibliographic record
Abstract
Working in the setting of i.i.d. last-passage percolation on R^D with no assumptions on the underlying edge-weight distribution, we develop the notion of grid entropy: a deterministic directed norm with negative sign that measures the proportion of empirical measures of edge weights (in a fixed direction or direction-free) which converge weakly to a given target measure. Though grid entropy and its convex duality to point-to-point/point-to-level Gibbs Free Energy have already been discovered by Rassoul-Agha and Seppalainen [19], our approach is novel in that we realize grid entropy as both a Subadditive Ergodic Theorem limit and equivalently as the threshold exponent of canonical order statistics associated with the Levy-Prokhorov metric. We use this new framework to re-derive various properties of grid entropy, including an upper bound on the sum of relative and grid entropies and upper semicontinuity. We also show that the direction-free case is nothing more than the direction-fixed case in the (1,1, ...,1) direction. In addition, we connect these results to the work of Bates [3] and partially answer a directed polymer version of a question of Hoffman. Shifting gears, we proceed to study these objects in a model consisting of repeatedly taking D samples from a distribution and picking out one according to an omniscient ”strategy.” We show that the set of limit points of empirical measures is almost surely the same whether or not we restrict ourselves to strategies which make the choices independently of all past and future choices, and moreover, that this set coincides with the set of measures with finite grid entropy. These setsare convex and weakly compact; we characterize their extreme points as those given by a natural ”greedy” deterministic strategy and we compute their grid entropy to be 0. This yields a description of the set of limit points of empirical measures as the closed convex hull of measures given by a density which is D Beta(1,D) distributed. We also derive a simplified version of a grid entropy-based variational formula for Gibbs Free Energy for this model, and we present the dual formula for grid entropy.
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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 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it