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 1Ω, the total resistance Rn between the root vertex and a busbar connecting all the vertices at the nth level is considered. A dynamical system is presented which approximates Rn, it is shown that the mean value 〈Rn〉 for this system approaches (1 + p)/(1 − p) as n → ∞, the distribution of Rn at large n is also examined. Additionally, a random sequence construction akin to a random Fibonacci sequence is used to approximate Rn; this sequence is shown to be related to the Legendre polynomials and its mean is shown to converge with |〈Rn〉 − (1 + p)/(1 − p)| ∼ n−1/2.
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