Open Access
June 2009 Effective resistance of random trees
Louigi Addario-Berry, Nicolas Broutin, Gábor Lugosi
Ann. Appl. Probab. 19(3): 1092-1107 (June 2009). DOI: 10.1214/08-AAP572


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=2dXe 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)ln n+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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Louigi Addario-Berry. Nicolas Broutin. Gábor Lugosi. "Effective resistance of random trees." Ann. Appl. Probab. 19 (3) 1092 - 1107, June 2009.


Published: June 2009
First available in Project Euclid: 15 June 2009

zbMATH: 1176.60068
MathSciNet: MR2537200
Digital Object Identifier: 10.1214/08-AAP572

Primary: 60J45
Secondary: 31C20

Keywords: Efron–Stein inequality , Electrical networks , Random trees

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.19 • No. 3 • June 2009
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