Percolation Problems on N -Ary Trees

Tianxiang Ren () and Jinwen Wu
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Tianxiang Ren: Institute of Management, University of Science and Technology of China, Heifei 230026, China
Jinwen Wu: Institute of Management, University of Science and Technology of China, Heifei 230026, China

Mathematics, 2023, vol. 11, issue 11, 1-14

Abstract: 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.

Keywords: site percolation; accessibility percolation; head run; increasing path; N -ary tree (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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