@article{EJP414,
author = {Fabio Lucio Toninelli},
title = {Correlation Lengths for Random Polymer Models and for Some Renewal Sequences},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {12},
year = {2007},
keywords = {Pinning and Wetting Models, Typical and Average Correlation Lengths, Critical Exponents, Renewal Theory, Exponential Convergence Rates},
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.},
pages = {no. 21, 613-636},
issn = {1083-6489},
doi = {10.1214/EJP.v12-414},
url = {http://ejp.ejpecp.org/article/view/414}}