Noise-stability and central limit theorems for effective resistance of random electric networks

• Rossignol, Raphaël. ^[1]
  1. [1] Grenoble Alpes University

     Grenoble Alpes University

     Arrondissement de Grenoble, Francia

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 2, 2016, págs. 1053-1106
• Idioma: inglés
• DOI: 10.1214/14-AOP996
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• Resumen

  • We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective resistances are uniformly stable to noise. For graphs that satisfy some homogeneity property, we show in addition that it is concentrated on sets of small diameter. As a consequence, we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length n in Zd, when n goes to infinity.