Percolation Problems on IN/I-Ary Trees

Percolation theory is a subject that has been flourishing in recent decades. Because of its simple expression and rich connotation, it is widely used in chemistry, ecology, physics, materials science, infectious diseases, and complex networks. Consider an infinite-rooted N-ary tree where each vertex...

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Bibliographic Details
Published inMathematics (Basel) Vol. 11; no. 11
Main Authors Ren, Tianxiang, Wu, Jinwen
Format Journal Article
LanguageEnglish
Published MDPI AG 01.06.2023
Subjects
Analysis
Trees (Graph theory)
China
Online AccessGet full text
ISSN2227-7390
2227-7390
DOI10.3390/math11112571

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Summary:Percolation theory is a subject that has been flourishing in recent decades. Because of its simple expression and rich connotation, it is widely used in chemistry, ecology, physics, materials science, infectious diseases, and complex networks. Consider an infinite-rooted N-ary tree where each vertex is assigned an i.i.d. random variable. When the random variable follows a Bernoulli distribution, a path is called head run if all the random variables that are assigned on the path are 1. We obtain the weak law of large numbers for the length of the longest head run. In addition, when the random variable follows a continuous distribution, a path is called an increasing path if the sequence of random variables on the path is increasing. By Stein’s method and other probabilistic methods, we prove that the length of the longest increasing path with a probability of one focuses on three points. We also consider limiting behaviours for the longest increasing path in a special tree.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11112571