First-Passage Percolation on Spread-out line graphs: Microscopic Regime

Abstract

We study first-passage percolation on the ℓ-spread-out line graph, where each vertex i∈{0,1,…,n} is connected to all others at distance at most ℓ. Here, we focus on the microscopic regime, with ℓ fixed as n→∞. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time T_n from $0$ to n. If the weight distribution has finite variance or a heavy tail with exponent above 2/ℓ_c where ℓ_c=ℓ(ℓ+1)/2, then T_n satisfies a Gaussian CLT with √n scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that T_n, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of T_n.

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BibTeX RIS

BibTeX

@misc{dey2026firstpassage,
  title = {{First-Passage Percolation on Spread-out line graphs: Microscopic Regime}},
  author = {Partha S. Dey and Daecheol Kim},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.28865},
  eprint = {2607.28865},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.28865},
}

RIS

TY  - UNPB
TI  - First-Passage Percolation on Spread-out line graphs: Microscopic Regime
AU  - Partha S. Dey
AU  - Daecheol Kim
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2607.28865
UR  - https://arxiv.org/abs/2607.28865
AB  - We study first-passage percolation on the ℓ-spread-out line graph, where each vertex i∈{0,1,…,n} is connected to all others at distance at most ℓ. Here, we focus on the microscopic regime, with ℓ fixed as n→∞. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time T_n from $0$ to n. If the weight distribution has finite variance or a heavy tail with exponent above 2/ℓ_c where ℓ_c=ℓ(ℓ+1)/2, then T_n satisfies a Gaussian CLT with √n scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that T_n, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of T_n.
ER  -

Abstract

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