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arXiv · 0801.1909

Effective resistance of random trees

Louigi Addario-Berry·Nicolas Broutin·Gábor Lugosi
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Abstract

We investigate the effective resistance $R_n$ and conductance $C_n$ 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 $r_e=2^dX_e$ where the $X_e$ are i.i.d. positive random variables bounded away from zero and infinity. It is shown that $\mathbf{E}R_n=n\mathbf{E}X_e-(\operatorname {\mathbf{Var}}(X_e)/\mathbf{E}X_e)\ln n+O(1)$ and $\operatorname {\mathbf{Var}}(R_n)=O(1)$. Moreover, we establish sub-Gaussian tail bounds for $R_n$. We also discuss some possible extensions to supercritical Galton--Watson trees.

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Louigi Addario-Berry, Nicolas Broutin, Gábor Lugosi. 2009-08-07. Effective resistance of random trees. https://doi.org/10.1214/08-aap572

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