Limit theorems for patterns in ranked tree‐child networks

We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree‐child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (Random Struct. Algoritm. 60 (2022), no. 4, 653–689). For patterns of height 1 and...

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Bibliographic Details
Published inRandom structures & algorithms Vol. 64; no. 1; pp. 15 - 37
Main Authors Fuchs, Michael, Liu, Hexuan, Yu, Tsan‐Cheng
Format Journal Article
LanguageEnglish
Published New York John Wiley & Sons, Inc 01.01.2024
Wiley Subscription Services, Inc
Subjects
Asymptotic properties
Cherries
Diffraction patterns
limit law
method of moments
pattern
Phylogenetic network
ranked tree‐child network
tree‐child network
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Summary:We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree‐child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (Random Struct. Algoritm. 60 (2022), no. 4, 653–689). For patterns of height 1 and 2, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to 0 and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern.
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21177