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Disorder relevance at marginality and critical point shift. (English. French summary) Zbl 1210.82036
Summary: Recently the renormalization group predictions on the effect of disorder on pinning models have been put on mathematical grounds. The picture is particularly complete if the disorder is relevant or irrelevant in the Harris criterion sense: the question addressed is whether quenched disorder leads to a critical behavior which is different from the one observed in the pure, i.e. annealed, system. The Harris criterion prediction is based on the sign of the specific heat exponent of the pure system, but it yields no prediction in the case of vanishing exponent. This case is called marginal, and the physical literature is divided on what one should observe for marginal disorder, notably there is no agreement on whether a small amount of disorder leads or not to a difference between the critical point of the quenched system and the one for the pure system. In [Commun. Pure Appl. Math. 63, No. 2, 233–265 (2010; Zbl 1189.60173)] we have proven that the two critical points differ at marginality of at least $$\exp(-c/\beta^4$$), where $$c>0$$ and $$\beta^2$$ is the disorder variance, for $$\beta \in (0, 1)$$ and Gaussian IID disorder. The purpose of this paper is to improve such a result: we establish in particular that the $$\exp(-c/\beta^4$$) lower bound on the shift can be replaced by $$\exp(-c(b)/\beta^b)$$, $$c(b)>0$$ for $$b>2$$ $$(b=2$$ is the known upper bound and it is the result claimed in [B. Derrida, V. Hakim and J. Vannimenus, J. Stat. Phys. 66, No. 5–6, 1189–1213 (1992; Zbl 0900.82051)]), and we deal with very general distribution of the IID disorder variables. The proof relies on coarse graining estimates and on a fractional moment change of measure argument based on multi-body potential modifications of the law of the disorder.

##### MSC:
 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B27 Critical phenomena in equilibrium statistical mechanics 60K37 Processes in random environments
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