Universality in the mean spatial shape of avalanches

Quantifying the universality of avalanche observables beyond critical exponents is of current great interest in theory and experiments. Here, we improve the characterization of the spatio-temporal process inside avalanches in the universality class of the depinning of elastic interfaces in random media. Surprisingly, at variance with the temporal shape, the spatial shape of avalanches has not yet been predicted. In part this is due to a lack of an analytically tractable definition: how should the shapes be centered? Here we introduce such a definition, accessible in experiments, and study the mean spatial shape of avalanches at fixed size centered around their starting point (seed). We calculate the associated universal scaling functions, both in a mean-field model and beyond. Notably, they are predicted to exhibit a cusp singularity near the seed. The results are in good agreement with a numerical simulation of an elastic line.