Avalanches and dimensional reduction breakdown in the critical behavior of disordered systems

TL;DR

This paper explores how avalanches cause a breakdown of the dimensional reduction property in disordered systems at criticality, linking nonanalyticities in disorder cumulants to the fractal nature of avalanches.

Contribution

It establishes a precise condition relating avalanche fractal dimension to the breakdown of dimensional reduction using scaling theory and functional renormalization group.

Findings

Dimensional reduction fails when avalanche fractal dimension equals the difference between space dimension and scaling dimension.

Dimensional reduction remains valid in random field systems above a critical dimension.

Dimensional reduction is always valid for branched polymers and invalid in elastic interface models.

Abstract

We investigate the connection between a formal property of the critical behavior of several systems in the presence of quenched disorder, known as "dimensional reduction", and the presence in the same systems at zero temperature of collective events known as "avalanches". Avalanches generically produce nonanalyticities in the functional dependence of the cumulants of the renormalized disorder. We show that this leads to a breakdown of the dimensional reduction predictions if and only if the fractal dimension characterizing the scaling properties of the avalanches is exactly equal to the difference between the dimension of space and the scaling dimension of the primary field, e.g. the magnetization in a random field model. This is proven by combining scaling theory and functional renormalization group. We therefore clarify the puzzle of why dimensional reduction remains valid in random field systems above a nontrivial dimension (but fails below), always applies to the statistics of branched polymer and is always wrong in elastic models of interfaces in a random environment.