Open Access
2017 Disorder relevance without Harris Criterion: the case of pinning model with $\gamma $-stable environment
Hubert Lacoin, Julien Sohier
Electron. J. Probab. 22: 1-26 (2017). DOI: 10.1214/17-EJP66


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.


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Hubert Lacoin. Julien Sohier. "Disorder relevance without Harris Criterion: the case of pinning model with $\gamma $-stable environment." Electron. J. Probab. 22 1 - 26, 2017.


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

zbMATH: 1368.60104
MathSciNet: MR3666013
Digital Object Identifier: 10.1214/17-EJP66

Primary: 60K35 , 60K37 , 82B27 , 82B44

Keywords: disorder relevance , Harris criterion , pinning model , Stable laws

Vol.22 • 2017
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