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2008 On the Approach to Equilibrium for a Polymer with Adsorption and Repulsion
Pietro Caputo, Fabio Martinelli, Fabio Toninelli
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Electron. J. Probab. 13: 213-258 (2008). DOI: 10.1214/EJP.v13-486


We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after $L$ steps, $L\in 2\mathbb{N}$. In the <em>pinning model</em> 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 %associated to 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 (\ie $\lambda=1$ without the 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 $\lambda\geq 1$ 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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Pietro Caputo. Fabio Martinelli. Fabio Toninelli. "On the Approach to Equilibrium for a Polymer with Adsorption and Repulsion." Electron. J. Probab. 13 213 - 258, 2008.


Accepted: 22 February 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1191.60110
MathSciNet: MR2386733
Digital Object Identifier: 10.1214/EJP.v13-486

Primary: 60K35
Secondary: 82C20

Keywords: coupling , dynamical phase transition , mixing time , pinning model , spectral gap


Vol.13 • 2008
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