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arXiv · 1703.03639

Critical percolation on random regular graphs

Felix Joos·Guillem Perarnau
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Abstract

We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known results for fixed $d\geq 3$ and for $d=n-1$, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random $d$-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method.

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Felix Joos, Guillem Perarnau. 2017-03-10. Critical percolation on random regular graphs. https://arxiv.org/abs/1703.03639

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