Metastability under stochastic dynamics

This paper is a tutorial introduction to some of the mathematics behind metastable behavior of interacting particle systems. The main focus is on the formation of so-called critical droplets, in particular, on their geometry and the time of their appearance. Special attention is given to Ising spins subject to a Glauber spin-flip dynamics and lattice particles subject to a Kawasaki hopping dynamics. The latter is one of the hardest models that can be treated to date and therefore is representative for the current state of development of this research area.