Correlation lengths for random polymer models and for some renewal sequences
Fabio Lucio Toninelli, ENS LYON
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
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 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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