The Resistance Of Randomly Grown Trees

Abstract

An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability $p$ or two edges with probability $1-p$. With each edge having a resistance equal to 1, the total resistance $R_n$ between the root vertex and a busbar connecting all the vertices at the $n^{th}$ level is considered. Representing $R_n$ as a dynamical system it is shown that $⟨R_n⟩$ approaches $(1+p)/(1-p)$ as $n\to \infty$, the distribution of $R_n$ at large $n$ is also examined. Additionally, expressing $R_n$ as a random sequence, its mean is shown to be related to the Legendre polynomials and that it converges to the mean with $|⟨R_n⟩-(1+p)/(1-p)|\sim n^{-1/2}$.

References in corpus (2)

• Random Fibonacci Sequences
• Growth and Structure of Stochastic Sequences