Osaka Journal of Mathematics

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

Yosuke Ota

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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.

Article information

Osaka J. Math., Volume 52, Number 3 (2015), 677-721.

First available in Project Euclid: 17 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability


Ota, Yosuke. The convergence of the exploration process for critical percolation on the $k$-out graph. Osaka J. Math. 52 (2015), no. 3, 677--721.

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