Electronic Journal of Probability

Disorder relevance without Harris Criterion: the case of pinning model with $\gamma$-stable environment

Abstract

We investigate disorder relevance for the pinning of a renewal whose inter-arrival law has tail exponent $\alpha >0$ when the law of the random environment is in the domain of attraction of a stable law with parameter $\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not decided by the sign of the specific heat exponent as predicted by Harris criterion but that a new criterion emerges to decide disorder relevance. More precisely we show that when $\alpha >1-\gamma ^{-1}$ there is a shift of the critical point at every temperature whereas when $\alpha < 1-\gamma ^{-1}$, at high temperature the quenched and annealed critical points coincide, and the critical exponents are identical.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 50, 26 pp.

Dates
Accepted: 4 May 2017
First available in Project Euclid: 13 June 2017

https://projecteuclid.org/euclid.ejp/1497319467

Digital Object Identifier
doi:10.1214/17-EJP66

Mathematical Reviews number (MathSciNet)
MR3666013

Zentralblatt MATH identifier
1368.60104

Citation

Lacoin, Hubert; Sohier, Julien. Disorder relevance without Harris Criterion: the case of pinning model with $\gamma$-stable environment. Electron. J. Probab. 22 (2017), paper no. 50, 26 pp. doi:10.1214/17-EJP66. https://projecteuclid.org/euclid.ejp/1497319467

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