The Moran forest. Issue 2 (15th February 2021)

• Record Type: Journal Article
• Authors: Bienvenu, François; Duchamps, Jean‐Jil; Foutel‐Rodier, Félix
• Is Part Of: Random structures & algorithms. Volume 59:Issue 2(2021)
• Page Start: 155
• Page End: 188
• Publication Date: 2021-02-15
• Subjects: dynamic graph -- Joyal bijection -- Moran model -- random forest -- random tree -- uniform tree; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20997
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: Starting from any graph on {1, … , n }, consider the Markov chain where at each time‐step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of nonempty forests on {1, … , n }. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit distribution of the size of a tree sampled uniformly. We also show that the size of the largest tree is asymptotically α log n, where α = ( 1 − log ( e − 1 ) ) − 1 ≈ 2 . 18, and that the degree of the most connected vertex is asymptotically log n / log log n .