Universality for the pinning model in the weak coupling regime

Abstract

We consider disordered pinning models, when the return time distribution of the underlying renewal process has a polynomial tail with exponent $\alpha \in (1/2,1)$. This corresponds to a regime where disorder is known to be relevant, i.e. to change the critical exponent of the localization transition and to induce a non-trivial shift of the critical point. We show that the free energy and critical curve have an explicit universal asymptotic behavior in the weak coupling regime, depending only on the tail of the return time distribution and not on finer details of the models. This is obtained comparing the partition functions with corresponding continuum quantities, through coarse-graining techniques.

arXiv comment36 pages, 1 figure

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• Scaling limits of disordered systems and disorder relevance
• Localization, big-jump regime and the effect disorder for a class of generalized pinning models
• Free energy of directed polymers in random environment in $1+1$-dimension at high temperature
• The critical disordered pinning measure
• Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime
• Scaling limit of the disordered generalized Poland--Scheraga model for DNA denaturation
• Two-sided bounds on free energy of directed polymers on strongly recurrent graphs