On the approach to equilibrium for a polymer with adsorption and repulsion
Pietro Caputo, Universita Roma Tre
Fabio Martinelli, Universita Roma Tre
Fabio Lucio Toninelli, ENS Lyon
Abstract
We consider paths of a one-dimensional simple random walk conditioned
to come back to the origin after L steps. In the
pinning model each path has a weight lambdaN, where
lambda>0 and N is the number of zeros in the path. 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 without the wall), where the
gap and the mixing time are known to scale as L-2 and L2 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 > 1 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.
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