### 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.

Original language | English |
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Pages (from-to) | 3477-3498 |

Number of pages | 22 |

Journal | Stochastic Processes and their Applications |

Volume | 130 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2020 |

### Keywords

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