On the approach to equilibrium for a polymer with adsorption and repulsion

TL;DR

This paper analyzes the dynamics of a one-dimensional polymer model with adsorption and repulsion, providing new bounds on relaxation times and spectral gaps, and revealing how phase transitions affect the system's approach to equilibrium.

Contribution

It introduces a natural spin flip dynamics for the polymer model and derives improved estimates on spectral gap and mixing time, highlighting the impact of phase transitions on relaxation behavior.

Findings

Relaxation to equilibrium is at least as fast as the free case with no wall.

In the delocalized phase, the spectral gap scales as O(L^{-5/2}) with logarithmic corrections.

Localized regime exhibits stretched exponential relaxation of local functions.

Abstract

We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path \eta has a weight \lambda^{N(\eta)}, where \lambda>0 and N(\eta) is the number of zeros in \eta. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength \lambda is varied. In this paper we study a natural 'spin flip' dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (\lambda=1, no wall), where the gap and the mixing time are known to scale as L^{-2} and L^2\log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for \lambda \geq 1 the relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (\lambda < 1) the gap is shown to be O(L^{-5/2}), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.