GAP TAILS AND FRAGMENTATION IN ONE DIMENSION: An exact reduction for Poisson and GUE proximity graphs

Q: What does this research mean for the field?

One-dimensional hard-radius proximity graphs can be exactly reduced to consecutive-gap exceedances, revealing a GUE-Poisson fragmentation crossover near one mean spacing. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

Thomas Tripi

doi:10.5281/zenodo.22165186

August 29, 2026Open Access

GAP TAILS AND FRAGMENTATION IN ONE DIMENSION: An exact reduction for Poisson and GUE proximity graphs

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Key Points

Determine exact conditions under which one-dimensional hard-radius proximity graphs undergo fragmentation using gap statistics from Poisson and Gaussian Unitary Ensemble point processes.
Reduced the connectivity of one-dimensional proximity graphs mathematically to consecutive-gap exceedance probabilities.
Derived analytical benchmark solutions for Poisson point processes and conducted numerical simulations using Gaussian Unitary Ensemble (GUE) random matrices.
Identified an exact crossover between GUE and Poisson fragmentation behavior occurring near one mean spacing unit.
Demonstrated numerically that a two-regime structure governs the graph, establishing that local fragmentation decouples from global graph connectivity.

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Implication

Theoretical analysis reveals a fragmentation crossover in one-dimensional proximity graphs, indicating distinct mechanisms separating local fragmentation from whole-graph connectivity.

Abstract

Exact reduction of one-dimensional hard-radius proximity graphs to consecutive-gap exceedances. The result gives closed Poisson benchmarks and identifies a GUE–Poisson fragmentation crossover near one mean spacing. Numerical GUE experiments confirm the two-regime structure and separate local fragmentation from whole-graph connectivity.

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Thomas Tripi (2026) studied this question.

synapsesocial.com/papers/6a93f1266c1a8fb52e79df39 https://doi.org/10.5281/zenodo.22165186

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GAP TAILS AND FRAGMENTATION IN ONE DIMENSION: An exact reduction for Poisson and GUE proximity graphs

Key Points

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Implication

Abstract

Discussion

Cite This Study

GAP TAILS AND FRAGMENTATION IN ONE DIMENSION: An exact reduction for Poisson and GUE proximity graphs

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

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