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Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimension.
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Through the NP-FRG we have also studied the RFIM with long-ranged interactions and/or correlations of the random field and found a similar scenario: I balog, G. tarjus, and M. Tissier, J. Stat. Mech. P10017 (2014); M. Baczyk, M. Tissier, G. Tarjus, and Y Sakamoto, Phys. Rev. B 88
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From scaling arguments one can attribute to the exponent λ \lambda the following physical interpretation: it characterizes the divergence of the number of (critical) spanning avalanches at criticality, i.e
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M. Baczyk, G. Tarjus, M. Tissier, and I. Balog, J. Stat. Mech. P06010 (2014)
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