Correlation Lengths for Random Polymer Models and for Some Renewal Sequences

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 ξ, defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than 1/f. We prove a lower bound also for the rate of exponential decay of the disorder-averaged two-point function. 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. 2000 Mathematics Subject Classification: 82B27, 82B44, 82B41, 60K05