Tail Bounds for the Stable Marriage of Poisson and Lebesgue
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Bibliographic record
Abstract
Abstract Let 𝚵 be a discrete set in ℝ d . Call the elements of 𝚵 centers . The well-known Voronoi tessellation partitions ℝ d into polyhedral regions (of varying volumes) by allocating each site of ℝ d to the closest center. Here we study allocations of ℝ d to 𝚵 in which each center attempts to claima region of equal volume α. We focus on the case where 𝚵 arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance X from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite α. In the critical case α = 1 we prove a power law upper bound on X in dimension d = 1. (Power law lower bounds were proved earlier for all d ). In the non-critical cases α < 1 and α > 1 we prove exponential upper bounds on X .
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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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| 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 |
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