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Avalanche size distributions in mean field plastic yielding models

I discuss the size distribution ${\cal N}(S)$ of avalanches occurring at the yielding transition of mean field (i.e., Hebraud-Lequeux) models of amorphous solids. The size distribution follows a power law dependence of the form: ${\cal N}(S)\sim S^{-τ}$. However (contrary to what is found in its depinning counterpart) the value of $τ$ depends on details of the dynamic protocol used. For random triggering of avalanches I recover the $τ=3/2$ exponent typical of mean field models, which in particular is valid for the depinning case. However, for the physically relevant case of external loading through a quasistatic increase of applied strain, a smaller exponent (close to 1) is obtained. This result is rationalized by mapping the problem to an effective random walk in the presence of a moving absorbing boundary.