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2007 Correlation Lengths for Random Polymer Models and for Some Renewal Sequences
Fabio Lucio Toninelli
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Electron. J. Probab. 12: 613-636 (2007). DOI: 10.1214/EJP.v12-414


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.


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Fabio Lucio Toninelli. "Correlation Lengths for Random Polymer Models and for Some Renewal Sequences." Electron. J. Probab. 12 613 - 636, 2007.


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

zbMATH: 1136.82013
MathSciNet: MR2318404
Digital Object Identifier: 10.1214/EJP.v12-414

Primary: 82B27
Secondary: 60K05 , 82B41 , 82B44

Keywords: Critical exponents , Exponential Convergence Rates , Pinning and wetting models , renewal theory , Typical and Average Correlation Lengths

Vol.12 • 2007
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