A down‐up chain with persistent labels on multifurcating trees

IF 0.9 3区数学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Random Structures & Algorithms Pub Date : 2023-10-12 DOI:10.1002/rsa.21185

Frederik Sørensen

{"title":"A down‐up chain with persistent labels on multifurcating trees","authors":"Frederik Sørensen","doi":"10.1002/rsa.21185","DOIUrl":null,"url":null,"abstract":"Abstract In this article, we propose to study a general notion of a down‐up Markov chain for multifurcating trees with labeled leaves. We study in detail down‐up chains associated with the ‐model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018; Random Struct. Algoritm. 54 (2020), 745–769; Electron. J. Probab. 25 (2020), 1–46.) in the binary special cases. The technique we deploy utilizes the construction of a growth process and a down‐up Markov chain on trees with planar structure. Our construction ensures that natural projections of the down‐up chain are Markov chains in their own right. We establish label dynamics that at the same time preserve the labeled alpha‐gamma distribution and keep the branch points between the smallest labels for order time steps for all . We conjecture the existence of diffusive scaling limits generalizing the “Aldous diffusion” by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018.) as a continuum‐tree‐valued process and the “algebraic ‐Ford tree evolution” by Löhr et al. (Ann. Probab. 48 (2020), 2565–2590.) and by Nussbaumer and Winter (arXiv:2006.09316, 2020.) as a process in a space of algebraic trees.","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"33 1","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21185","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}

引用次数: 1

Abstract

Abstract In this article, we propose to study a general notion of a down‐up Markov chain for multifurcating trees with labeled leaves. We study in detail down‐up chains associated with the ‐model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018; Random Struct. Algoritm. 54 (2020), 745–769; Electron. J. Probab. 25 (2020), 1–46.) in the binary special cases. The technique we deploy utilizes the construction of a growth process and a down‐up Markov chain on trees with planar structure. Our construction ensures that natural projections of the down‐up chain are Markov chains in their own right. We establish label dynamics that at the same time preserve the labeled alpha‐gamma distribution and keep the branch points between the smallest labels for order time steps for all . We conjecture the existence of diffusive scaling limits generalizing the “Aldous diffusion” by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018.) as a continuum‐tree‐valued process and the “algebraic ‐Ford tree evolution” by Löhr et al. (Ann. Probab. 48 (2020), 2565–2590.) and by Nussbaumer and Winter (arXiv:2006.09316, 2020.) as a process in a space of algebraic trees.

查看原文本刊更多论文

在多分形树上具有持久标签的自上而下链

摘要本文研究了带有标记叶的多分形树的下向上马尔可夫链的一般概念。我们详细研究了与Chen等人的模型相关的上下链。J. Probab. 14(2009)， 400-430 .)，对Forman等人先前工作的推广和进一步发展(arXiv:1802.00862, 2018;arXiv: 1804.01205, 2018;arXiv: 1809.07756, 2018;随机结构。算法，54 (2020)，745-769;电子。[j] .概率学报，25 (2020)，1-46 .]我们采用的技术是在平面结构的树上构造一个生长过程和一个自上而下的马尔可夫链。我们的构造保证了上下链的自然投影本身就是马尔可夫链。我们建立了标签动力学，同时保留标记的alpha - gamma分布，并在所有的阶时间步长中保持最小标签之间的分支点。本文通过Forman等人的推广，推测了扩散标度极限的存在性。[j] .数学学报:自然科学版，2018;arXiv: 1804.01205, 2018;(Ann.)作为连续统-树值过程和“代数-福特树进化”的Löhr等人。(Probab. 48(2020)， 2565-2590 .)和Nussbaumer and Winter (arXiv:2006.09316, 2020.)作为代数树空间中的过程。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Random Structures & Algorithms 数学-计算机：软件工程

CiteScore

2.50

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness. Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.