Percolation Problems on N-Ary Trees

• Published in: Mathematics (Basel) Vol. 11; no. 11; p. 2571
• Main Authors: Ren, Tianxiang, Wu, Jinwen
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 04.06.2023
• Subjects: accessibility percolation; Continuity (mathematics); Evolutionary biology; Genotype & phenotype; Graphs; head run; increasing path; Infectious diseases; Materials science; N-ary tree; Percolation theory; Probabilistic methods; Probability theory; Random variables; Securities prices; site percolation; Star & galaxy formation; Statistical analysis; Trees; Trees (mathematics)
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11112571

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.