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Effective resistance of random trees. (English) Zbl 1176.60068
The authors consider a binary tree of height $$n$$ and investigate the effective resistance $$R_n$$ and conductance $$C_n$$ between its root and leaves. In this electrical network, the resistance of each edge $$e$$ at distance $$d$$ from the root is defined by $$r_e := 2^d X_e$$, where the $$X_e$$ are i. i. d. positive random variables, bounded away from zero at infinity. It is shown that $$\text{E} R_n = n \text{E} X_e - ( \text{Var} (X_e)/ \text{E} X_e ) \ln n + O(1)$$ and $$\text{Var} (R_n) = O(1)$$. Moreover, they establish sub–Gaussian tail bounds for $$R_n$$. Some possible extensions to supercritical Galton–Watson trees are also discussed.

##### MSC:
 60J45 Probabilistic potential theory 31C20 Discrete potential theory
##### Keywords:
random trees; electrical networks; Efron–Stein inequality
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##### References:
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