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 Levy 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 -> 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 which it has positive or weakly zero speed when the mean of the largest conductance is finite or infinite, respectively.
Publisher
ELSEVIER
Issue Date
2020-06
Language
English
Article Type
Article
Citation

STOCHASTIC PROCESSES AND THEIR APPLICATIONS, v.130, no.6, pp.3477 - 3498

ISSN
0304-4149
DOI
10.1016/j.spa.2019.10.001
URI
http://hdl.handle.net/10203/274230
Appears in Collection
MA-Journal Papers(저널논문)
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