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

Author

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  • Collevecchio, Andrea
  • Jung, Paul

Abstract

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.

Suggested Citation

  • Collevecchio, Andrea & Jung, Paul, 2020. "On the speed and spectrum of mean-field random walks among random conductances," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3477-3498.
  • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3477-3498
    DOI: 10.1016/j.spa.2019.10.001
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    References listed on IDEAS

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    1. Gantert, Nina & Müller, Sebastian & Popov, Serguei & Vachkovskaia, Marina, 2012. "Random walks on Galton–Watson trees with random conductances," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1652-1671.
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