A mathematical perspective on metastable wetting

Abstract

In this paper we investigate the dynamical behavior of an interface or polymer, in interaction with a distant attractive substrate. The interface is modeled by the graph of a nearest neighbor path with non-negative integer coordinates, and the equilibrium measure associates to each path $\eta$ a probability proportional to $\lambda^{H(\eta)}$ where $\lambda$ is non-negative and $H(\eta)$ is the number of contacts between $\eta$ and the substrate at zero. The dynamics is the natural "spin flip" dynamics associated to this equilibrium measure. We let the distance to the substrate at both polymer ends be equal to $aN$, where $a \in (0,1/2)$ is a fixed parameter, and $N$ is the length the system. With this setup, we show that the dynamical behavior of the system crucially depends on $\lambda$: when $\lambda \leq 2/(1-2a)$ we show that the system only needs a time which is polynomial in $N$ to reach its equilibrium state, whereas if $\lambda > 2/(1-2a)$ the mixing time is exponential in $N$ and the system relaxes in an exponential manner which is typical of metastability.