Electronic Journal of Probability

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

Hubert Lacoin and Julien Sohier

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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

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

Received: 21 October 2016
Accepted: 4 May 2017
First available in Project Euclid: 13 June 2017

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82B27: Critical phenomena 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Pinning model disorder relevance stable laws Harris criterion

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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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