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Submean variance bound for effective resistance of random electric networks. (English) Zbl 1207.82024

Electric networks on a given graph can be set in correspondence with a reversible Markov chain on the same graph. The randomness is introduced in an electric network by choosing independent and indentically distributed (electric) resistances. Next, the effective resistance is investigated between two finite sets of vertices. The focus of the present paper is on point-to-point effective resistance, the notion appearing in the first passage percolation problems. The main result of the paper is the variance bound for resistances distributed according to a Bernoulli distribution. The main technical tool is a modified Poincaré inequality, first introduced by D. Falik and A. Samorodnitsky [Edge-isoperimetric inequalities and influences. arxiv:math.CO/0512636 (2005)].

MSC:

82B43 Percolation
05C80 Random graphs (graph-theoretic aspects)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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