Abstract
Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p. The method from Ajtai, Komlós and Szemerédi [Combinatorica 2 (1982) 1–7] is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth, as well as a simple proof of a result of Kesten about the critical probability for bond percolation in high dimensions. Several problems and conjectures regarding percolation on finite transitive graphs are presented.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
