## Electronic Journal of Probability

### On the Approach to Equilibrium for a Polymer with Adsorption and Repulsion

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 10, 213-258.

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

https://projecteuclid.org/euclid.ejp/1464819083

Digital Object Identifier
doi:10.1214/EJP.v13-486

Mathematical Reviews number (MathSciNet)
MR2386733

Zentralblatt MATH identifier
1191.60110

Rights

#### Citation

Caputo, Pietro; Martinelli, Fabio; Toninelli, Fabio. On the Approach to Equilibrium for a Polymer with Adsorption and Repulsion. Electron. J. Probab. 13 (2008), paper no. 10, 213--258. doi:10.1214/EJP.v13-486. https://projecteuclid.org/euclid.ejp/1464819083

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