Avalanche-size distributions in mean-field plastic yielding models

We discuss the size distribution 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 N(S) similar to S-tau. However (contrary to what is found in its depinning counterpart), the value of tau depends on details of the dynamic protocol used. For random triggering of avalanches we recover the tau = 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.