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arXiv · 1702.04087

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

Andrea Collevecchio·Paul Jung
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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 L\'evy 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 goes to infinity, 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 that it has positive or weakly zero speed when the mean of the largest conductance is finite or infinite, respectively.

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Andrea Collevecchio, Paul Jung. 2017-02-14. On the speed and spectrum of mean-field random walks among random conductances. https://arxiv.org/abs/1702.04087

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