## Electronic Journal of Probability

### Correlation Lengths for Random Polymer Models and for Some Renewal Sequences

Fabio Lucio Toninelli

#### 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

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

Rights

#### 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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