MUTATIONS ON A RANDOM BINARY TREE WITH MEASURED BOUNDARY

• 成果类型: Article
• 署名作者: Duchamps, Jean-Jil; Lambert, Amaury
• 署名单位: Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Institut National de la Sante et de la Recherche Medicale (Inserm); Centre National de la Recherche Scientifique (CNRS); Universite PSL; College de France
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1353
• 发表日期: 2018
• 页码: 2141-2187
• 关键词: splitting trees

摘要

Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure (infinitely-many-allele model) on the skeleton of the tree. The partition of the boundary into distinct alleles is the so-called allelic partition. In this paper, we are interested in the infinite trees generated by supercritical, possibly time-inhomogeneous, binary branching processes, and in their boundary, which is the set of particles coexisting at infinity. We prove that any such tree can be mapped to a random, compact ultrametric tree called the coalescent point process, endowed with a uniform measure on its boundary which is the limit as t -> infinity of the properly rescaled counting measure of the population at time t. We prove that the clonal (i.e., carrying the same allele as the root) part of the boundary is a regenerative set that we characterize. We then study the allelic partition of the boundary through the measures of its blocks. We also study the dynamics of the clonal subtree, which is a Markovian increasing tree process as mutations are removed.