Abstract
Let be a fixed integer, , and let be a positive integer such that is even. Let be a (random) graph on n vertices obtained by drawing uniformly at random a d-regular (simple) graph on and then performing independent p-bond percolation on it, i.e. we independently retain each edge with probability p and delete it with probability . Let be the size of the largest component in . We show that, when p is of the form for , and A is large,
This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than .
Funding Statement
Both authors would like to thank the Royal Society for their generous funding, of a PhD scholarship for UDA and a University Research Fellowship for MR.
Acknowledgments
We would like to thank an anonymous referee for several helpful suggestions and corrections.
Citation
Umberto De Ambroggio. Matthew I. Roberts. "The probability of unusually large components for critical percolation on random d-regular graphs." Electron. J. Probab. 28 1 - 55, 2023. https://doi.org/10.1214/23-EJP982
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