Universality for the pinning model in the weak coupling regime

TL;DR

This paper demonstrates that in disordered pinning models with polynomial tail return times, the free energy and critical curve exhibit universal asymptotic behavior in the weak coupling regime, depending solely on the tail exponent.

Contribution

It establishes the universality of the asymptotic behavior of free energy and critical curve for a class of disordered pinning models in the weak coupling regime.

Findings

Universal asymptotics depend only on the tail exponent of the return time distribution.

Disorder relevance leads to a non-trivial shift in the critical point.

Comparison with continuum models via coarse-graining techniques confirms universality.

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.