Phase Transition and Fluctuation Results for First-Passage Percolation on Spread-Out Cycle Graphs

We study first-passage percolation on the $\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $ω_e \sim \mathrm{Exp}(1)^{1/θ}$ to the edges for $θ>0$ fixe