We establish metastability in the sense of Lebowitz and Penrose under
practical and simple hypothesis for (families of) Markov chains on finite
configuration space in some asymptotic regime, including the case of
configuration space size going to infinity. By comparing restricted ensemble
and quasi-stationary measures, we study point-wise convergence velocity of
Yaglom limits and prove asymptotic exponential exit law. We introduce soft
measures as interpolation between restricted ensemble and quasi-stationary
measure to prove an asymptotic exponential transition law on a generally
different time scale. By using potential theoretic tools, we prove a new
general Poincar\'e inequality and give sharp estimates via two-sided
variational principles on relaxation time as well as mean exit time and
transition time. We also establish local thermalization on a shorter time scale
and give mixing time asymptotics up to a constant factor through a two-sided
variational principal. All our asymptotics are given with explicit quantitative
bounds on the corrective terms.Comment: 41 page
