Conductance of a subdiffusive random weighted tree
Abstract
We work on a GaltonWatson tree with random weights, in the socalled "subdiffusive" regime. We study the rate of decay of the conductance between the root and the $n$th level of the tree, as $n$ goes to infinity, by a mostly analytic method. It turns out the order of magnitude of the expectation of this conductance can be less than $1/n$ (in contrast with the results of AddarioBerryBroutinLugosi and ChenHuLin), depending on the value of the second zero of the characteristic function associated to the model. We also prove the almost sure (and in $L^p$ for some $p>1$) convergence of this conductance divided by its expectation towards the limit of the additive martingale.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 DOI:
 10.48550/arXiv.1905.00821
 arXiv:
 arXiv:1905.00821
 Bibcode:
 2019arXiv190500821R
 Keywords:

 Mathematics  Probability;
 60J80;
 60G50;
 60F25;
 60F15
 EPrint:
 24 pages, 2 figures