  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.
