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

## TL;DR

This paper investigates the spectral properties and speed of random walks among random conductances on large complete graphs, revealing phase transitions related to conductance distribution and analyzing the limiting behavior on infinite trees.

## Contribution

It establishes the weak convergence of spectral distributions and characterizes the phase transition in the walk's speed based on conductance distribution properties.

## Key findings

- Spectral distribution converges to a symmetric measure on [-1,1].
- The underlying graph converges to a Poisson Weighted Infinite Tree.
- The walk exhibits a phase transition in speed depending on conductance mean.

## 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.