The convergence of the exploration process for critical percolation on the \(k\)-out graph
type: Item
instance of: scholarly article
title: The convergence of the exploration process for critical percolation on the \(k\)-out graph
zbMATH Open document ID: 1326.05144
author: Yosuke Ota
published in: Osaka Journal of Mathematics
publication date: 2015-10-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1437137614
review text: This paper studies critical percolation on the \(k\)-out graph \(G_{\mathrm{out}}(n,k)\) of order \(n\) via the analysis of exploration process, which explores open clusters of \(G_{\mathrm{out}}(n,k)\). This graph is undirected, and it can be derived from a directed version, which is constructed by adding uniformly at random \(k\) out-going edges to each vertex. The percolation process is performed by keeping each edge open with probability \(p\) and closed with probability \(1-p\) independently. The critical probability is shown to be \(p_c=1/(k+\sqrt{k^2-k})\). Moreover, the size of the largest component is of order \(O(\log n)\) in the subcritical phase, and \(O(n)\) in the supercritical phase. Some results are reminiscent of the classical random graph \(G(n,p)\).
reviewed by: Yilun Shang
Mathematics Subject Classification ID: 05C80
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 6502590
zbMATH Keywords: critical percolation
zbMATH Keywords: \(k\)-out graph
zbMATH Keywords: random graph
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6502590