  The concept of metastability has caused a lot of interest in recent years.
The spectral decomposition of the generator matrix of a stochastic network
exposes all of the transition processes in the system. The assumption of the
existence of a low lying group of eigenvalues separated by a spectral gap,
leading to factorization of the dynamics, has become a popular theme. We
consider stochastic networks representing potential energy landscapes where the
states and the edges correspond to local minima and transition states
respectively, and the pairwise transition rates are given by the Arrhenuis
formula. Using the minimal spanning tree, we construct the asymptotics for
eigenvalues and eigenvectors of the generator matrix starting from the low
lying group. This construction gives rise to an efficient algorithm for
computing the asymptotic spectrum suitable for large and complex networks. We
apply it to Wales's Lennard-Jones-38 network with 71887 states and 119853 edges
where the underlying potential energy landscape has a double-funnel structure.
Our results demonstrate that the concept of metastability should be applied
with care to this system. In particular, for the full network, there is no
significant spectral gap separating the eigenvalue corresponding to the exit
from the wider and shallower icosahedral funnel at any reasonable temperature
range.
