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 m² 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⟩Sₘ²p(S/Sₘ) where Sₘ:=⟨S²⟩2⟨S⟩~m^-d-ζ is the scale of avalanches, and ζ the roughness exponent at the depinning transition. Measurement of the scaling function f(s):=s^τp(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 τ is found in good agreement with the conjecture τ=2-2/(d+ζ), recently confirmed to one loop via the FRG. The function $f(s)$ exhibits a shoulder and a stretched exponential decay at large s, ln f(s)~-s^δ, with δ≈7/6 in $d=1$. The function $f(s)$, universal ratios of moments, and the generating function ⟨e^λs⟩ are found in excellent agreement with the one-loop FRG predictions. The distribution of local avalanche sizes S_φ, i.e., of the jumps of a subspace of the manifold of dimension d_φ, is also computed and compared to our FRG predictions, and to the conjecture τ_φ=2-2/(d_φ+ζ).