We describe a directed avalanche model; a slowly unloading sandbox driven by
lowering a retaining wall. The directness of the dynamics allows us to
interpret the stable sand surfaces as world sheets of fluctuating interfaces in
one lower dimension. In our specific case, the interface growth dynamics
belongs to the Kardar-Parisi-Zhang (KPZ) universality class. We formulate
relations between the critical exponents of the various avalanche distributions
and those of the roughness of the growing interface. The nonlinear nature of
the underlying KPZ dynamics provides a nontrivial test of such generic exponent
relations. The numerical values of the avalanche exponents are close to the
conventional KPZ values, but differ sufficiently to warrant a detailed study of
whether avalanche correlated Monte Carlo sampling changes the scaling exponents
of KPZ interfaces. We demonstrate that the exponents remain unchanged, but that
the traces left on the surface by previous avalanches give rise to unusually
strong finite-size corrections to scaling. This type of slow convergence seems
intrinsic to avalanche dynamics.Comment: 13 pages, 13 figure