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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 10 open journal systems 


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.


Full text: PDF

Pages: 213-258

Published on: February 22, 2008
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Electronic Journal of Probability. ISSN: 1083-6489