Effective resistance of random trees
Louigi Addario-Berry, Nicolas Broutin and Gábor Lugosi
Annals of Applied Probability Volume 19, pp. 1092-1107, 2009.

Abstract

We investigate the effective resistance Rn and conductance Cn between the root and leaves of a binary tree of height n. In this electrical network, the resistance of each edge e at distance d from the root is defined by re = 2d Xe where the Xe are i.i.d. positive random variables bounded away from zero and infinity. It is shown that ERn = nEXe − (Var(Xe)/EXe)lnn + O(1) and Var(Rn) = O(1). Moreover, we establish sub-Gaussian tail bounds for Rn. We also discuss some possible extensions to supercritical Galton– Watson trees.

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