Trees with exponential height dependent weight

• 成果类型: Article
• 署名作者: Durhuus, Bergfinnur; Unel, Meltem
• 署名单位: University of Copenhagen
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01188-7
• 发表日期: 2023
• 页码: 999-1043
• 关键词: galton-watson trees; Scaling Limit; random-walk; quadrangulations; plane

摘要

We consider planar rooted random trees whose distribution is even for fixed height h and size N and whose height dependence is of exponential form e(-mu h). Defining the total weight for such trees of fixed size to be Z((mu)) (N) , we determine its asymptotic behaviour for large N, for arbitrary real values of mu. Based on this we identify the local limit of the corresponding probability measures and find a transition at mu = 0 from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for mu < 0 to the familiar quadratic growth at mu = 0 and to cubic growth for mu > 0.