Electronic Journal of Probability

Correlation Lengths for Random Polymer Models and for Some Renewal Sequences

Fabio Lucio Toninelli

Full-text: Open access

Abstract

We consider models of directed polymers interacting with a one-dimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on $Z$ and gives a random (site-dependent) reward or penalty to the occurrence of a renewal at any given point of $Z$. These models are known to undergo a delocalization-localization transition, and the free energy $F$ vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length $\xi$, defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than $1/F$. We prove also an exponentially decaying upper bound for the disorder-averaged two-point function, with a good control of the sub-exponential prefactor. We discuss how, in the particular case where disorder is absent, this result can be seen as a refinement of the classical renewal theorem, for a specific class of renewal sequences.

Article information

Source
Electron. J. Probab., Volume 12 (2007), paper no. 21, 613-636.

Dates
Accepted: 13 May 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464818492

Digital Object Identifier
doi:10.1214/EJP.v12-414

Mathematical Reviews number (MathSciNet)
MR2318404

Zentralblatt MATH identifier
1136.82013

Subjects
Primary: 82B27: Critical phenomena
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60K05: Renewal theory

Keywords
Pinning and Wetting Models Typical and Average Correlation Lengths Critical Exponents Renewal Theory Exponential Convergence Rates

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Toninelli, Fabio Lucio. Correlation Lengths for Random Polymer Models and for Some Renewal Sequences. Electron. J. Probab. 12 (2007), paper no. 21, 613--636. doi:10.1214/EJP.v12-414. https://projecteuclid.org/euclid.ejp/1464818492


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