The Annals of Probability

On the irrelevant disorder regime of pinning models

Giambattista Giacomin and Fabio Lucio Toninelli

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Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.

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Ann. Probab., Volume 37, Number 5 (2009), 1841-1875.

First available in Project Euclid: 21 September 2009

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60K05: Renewal theory 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Directed polymers pinning and wetting models renewal theory irrelevant disorder Harris criterion intersection of renewals


Giacomin, Giambattista; Toninelli, Fabio Lucio. On the irrelevant disorder regime of pinning models. Ann. Probab. 37 (2009), no. 5, 1841--1875. doi:10.1214/09-AOP454.

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