A. Bovier, M. Eckhoff, V. Gayrard, M. Klein
Metastability and low lying spectra in reversible Markov chains
(232K, PS)
ABSTRACT. We study a large class of reversible
Markov chains with
discrete state space and transition matrix $P_N$.
We define the notion of a set of {\it metastable points}
as a subset of the state space $\G_N$ such that (i) this set is reached
from any point $x\in \G_N$ without return to $x$ with probability
at least $b_N$, while (ii) for any two point $x,y$ in the metastable set, the
probability $T^{-1}_{x,y}$ to reach $y$ from $x$ without return to $x$
is smaller than $a_N^{-1}\ll b_N$. Under some additional
non-degeneracy assumption, we show that in such a situation:
\item{(i)} To each metastable point corresponds a metastable state,
whose mean exit time can be computed precisely.
\item{(ii)} To each metastable point corresponds one simple eigenvalue
of $1-P_N$ which is essentially equal to
the inverse mean exit time from this state.
%The corresponding eigenfunctions
%are close to the indicator function of the support of the metastable state.
Moreover, these results imply very sharp uniform control of the deviation
of the probability distribution of metastable exit times from the exponential
distribution.