Coalescence of geodesics in exactly solvable models of last passage percolation
Bibliographic record
Abstract
Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper, we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z2 with independent and identically distributed exponential weights on the vertices. Fix two points v1 = (0, 0) and v2 = (0, k2/3) for some k > 0, and consider the maximal paths Γ1 and Γ2 starting at v1 and v2, respectively, to the point (n, n) for n ≫ k. Our object of study is the point of coalescence, i.e., the point v ∈ Γ1 ∩ Γ2 with smallest |v|1. We establish that the distance to coalescence |v|1 scales as k, by showing the upper tail bound P(|v|1>Rk)≤R−c for some c > 0. We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction (1, 1) starting from v3 = (−k2/3, k2/3) and v4 = (k2/3, −k2/3), we establish the optimal tail estimate P(|v|1>Rk)≍R−2/3, for the point of coalescence v. This answers a question left open by Pimentel [Ann. Probab. 44(5), 3187–3206 (2016)] who proved the corresponding lower bound.
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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 | 0.002 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".