On the speed and spectrum of mean-field random walks among random conductances

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We study random walk among random conductance (RWRC) on complete graphs with n vertices. The conductances are i.i.d. and the sum of conductances emanating from a single vertex asymptotically has an infinitely divisible distribution corresponding to a Lévy subordinator with infinite mass at 0. We show that, under suitable conditions, the empirical spectral distribution of the random transition matrix associated to the RWRC converges weakly, as n→∞, to a symmetric deterministic measure on [−1,1], in probability with respect to the randomness of the conductances. In short time scales, the limiting underlying graph of the RWRC is a Poisson Weighted Infinite Tree, and we analyze the RWRC on this limiting tree. In particular, we show that the transient RWRC exhibits a phase transition in which it has positive or weakly zero speed when the mean of the largest conductance is finite or infinite, respectively.

Original languageEnglish
Pages (from-to)3477-3498
Number of pages22
JournalStochastic Processes and their Applications
Issue number6
Publication statusPublished - Jun 2020


  • Empirical spectral distribution
  • Poisson weighted infinite tree
  • Random conductance model
  • Rate of escape
  • Speed

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