Universal interface width distributions at the depinning threshold.

We compute the probability distribution of the interface width at the depinning threshold, using recent powerful algorithms. It confirms the universality classes found previously. In all cases, the distribution is surprisingly well approximated by a generalized Gaussian theory of independent modes which decay with a characteristic propagator G(q)=1/q(d+2zeta); zeta, the roughness exponent, is computed independently. A functional renormalization analysis explains this result and allows one to compute the small deviations, i.e., a universal kurtosis ratio, in agreement with numerics. We stress the importance of the Gaussian theory to interpret numerical data and experiments.