Scaling limit and universality for one-dimensional long-range first-passage metric in the linear-growth regime
Abstract
We prove universal Gaussian and stable fluctuation limits for one-dimensional long-range first-passage percolation with edge times |x−y|αωx,y. The positive i.i.d. weights satisfy ℙ(ω≤ t)=tθ+o(1) as t↓0 and 𝔼(ωζ)<∞ for some θ,ζ>0. For α>1+2/θ, the passage time T(0,n) has a positive deterministic linear growth rate μ. For α>1+3/θ, we establish a Gaussian central limit theorem with linear asymptotic variance, a functional version, and limit theorems for geodesic hop and cut point counts. For 1+2/θ<α<1+3/θ, assuming a regularly varying lower tail, we obtain a totally left-skewed stable limit with index β=θ(α−1)−1 and an explicit scale, together with convergence to a spectrally negative stable Lévy process in the Skorokhod M1 topology. The proof constructs i.i.d. blocks between hubs through which every crossing geodesic must pass. A spread-out comparison metric makes this regeneration available for every ζ>0, even when the edge law has infinite mean. The key estimate identifies the exact left-tail asymptotic of the centered block reward, generated by a single fast long edge. A monotone within-block approximation yields the functional stable limit.
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BibTeX
@misc{chatterjee2026scaling,
title = {{Scaling limit and universality for one-dimensional long-range first-passage metric in the linear-growth regime}},
author = {Shirshendu Chatterjee and Partha S. Dey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37180},
eprint = {2610.37180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37180},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scaling limit and universality for one-dimensional long-range first-passage metric in the linear-growth regime
AU - Shirshendu Chatterjee
AU - Partha S. Dey
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37180
UR - https://arxiv.org/abs/2610.37180
AB - We prove universal Gaussian and stable fluctuation limits for one-dimensional long-range first-passage percolation with edge times |x−y|^αω_x,y. The positive i.i.d. weights satisfy ℙ(ω≤ t)=t^θ+o(1) as t↓0 and 𝔼(ω^ζ)<∞ for some θ,ζ>0. For α>1+2/θ, the passage time T(0,n) has a positive deterministic linear growth rate μ. For α>1+3/θ, we establish a Gaussian central limit theorem with linear asymptotic variance, a functional version, and limit theorems for geodesic hop and cut point counts. For 1+2/θ<α<1+3/θ, assuming a regularly varying lower tail, we obtain a totally left-skewed stable limit with index β=θ(α−1)−1 and an explicit scale, together with convergence to a spectrally negative stable Lévy process in the Skorokhod M_1 topology. The proof constructs i.i.d. blocks between hubs through which every crossing geodesic must pass. A spread-out comparison metric makes this regeneration available for every ζ>0, even when the edge law has infinite mean. The key estimate identifies the exact left-tail asymptotic of the centered block reward, generated by a single fast long edge. A monotone within-block approximation yields the functional stable limit.
ER -