The convergence of the exploration process for critical percolation on the $k$-out graph

Abstract

We consider the percolation on the $k$-out graph $G_{\textup{out}}(n, k)$. The critical probability of it is $p_{c}=1/(k+\sqrt{k^{2}-k})$. Similarly to the random graph $G(n, p)$, in a scaling window $p_{c}(1+O(n^{-1/3}))$, the sequence of sizes of large components rescaled by $n^{-2/3}$ converges to the excursion lengths of a Brownian motion with some drift. Also, the size of the largest component is $O(\log n)$ in the subcritical phase, and $O(n)$ in the supercritical phase. The proof is based on the analysis of the exploration process.