Avalanche-size distribution at the depinning transition: A numerical test of the theory

We calculate numerically the sizes S of jumps (avalanches) between successively pinned configurations of an elastic line (d = 1) or interface (d = 2), pulled by a spring of (small) strength m(2) in a random-field landscape. We obtain strong evidence that the size distribution, away from the small-scale cutoff, takes the form P(S) = < S >/S(m)(2)p(S/S(m)) where S(m) := < S(2)>/2 < S > similar to m(-d-zeta) is the scale of avalanches, and zeta the roughness exponent at the depinning transition. Measurement of the scaling function f(s) := s(tau)p(s) is compared with the predictions from a recent Functional RG (FRG) calculation, both at mean-field and one-loop level. The avalanche-size exponent tau is found in good agreement with the conjecture tau=2 - 2/(d + zeta), recently confirmed to one loop via the FRG. The function f(s) exhibits a shoulder and a stretched exponential decay at large s, In f(s)similar to-s(delta), with delta approximate to 7/ 6 in d = 1. The function f(s), universal ratios of moments, and the generating function < e(lambda s)> are found in excellent agreement with the one-loop FRG predictions. The distribution of local avalanche sizes S(phi), i.e., of the jumps of a subspace of the manifold of dimension d(phi), is also computed and compared to our FRG predictions, and to the conjecture tau(phi) = 2 - 2/(d(phi) + zeta).