Scaling limit and universality for one-dimensional long-range first-passage metric in the linear-growth regime

Source abstract

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 $\mathbb{P}(ω\le t)=t^{θ+o(1)}$ as $t\downarrow0$ and $\mathbb{E}(ω^ζ) 0$. 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 $1+2/θ 0$, 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.