Abstract
Let d > 3 be a fixed integer, p e (0; 1), and let n > 1 be a positive integer such that dn is even. Let G(n; d; p) be a (random) graph on n vertices obtained by drawing uniformly at random a d-regular (simple) graph on [n] and then performing independent p-bond percolation on it, i.e. we independently retain each edge with probability p and delete it with probability 1 - p. Let |C max| be the size of the largest component in G(n; d; p). We show that, when p is of the form p = (d - 1)-1(1 + -n -1/3) for e R, 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 An 2/3.
Original language | English |
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Pages (from-to) | 1-55 |
Number of pages | 55 |
Journal | Electronic Journal of Probability |
Volume | 28 |
Early online date | 11 Jul 2023 |
DOIs | |
Publication status | Published - 11 Jul 2023 |
Acknowledgements
We would like to thank an anonymous referee for several helpful suggestions and corrections.Funding
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.
Funders | Funder number |
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Royal Society |
Keywords
- Component size
- Exploration process
- Percolation
- Random regular graph
ASJC Scopus subject areas
- General Mathematics