Correlation lengths for random polymer models and for some renewal sequences
arXiv:math/0611868
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 and gives a random (site-dependent) reward or penalty to the occurrence of a renewal at any given point of . These models are known to undergo a delocalization-localization transition, and the free energy $\tf$ vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length , defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than $ 1/\tf$. 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.
21 pages, 2 figures; v2: results generalized, few typos corrected