SCALING OF OVERHANGS IN (1+1)-DIMENSIONAL DIRECTED PROCESSES IN A GRADIENT

When a gradient is imposed in the control parameter of a directed process in a (1+1)-dimensional directed process, a front may occur. The position of this front evolves in steps of size one in the direction opposite to the gradient, and in steps of size equal to or larger than one in the other direction. The larger steps are due to overhangs. There is a critical value for the control parameter only if the first moment of the step size distribution in the direction of the gradient is not singular-i.e. it takes on a finite value in the 'thermodynamic' limit. If there is a critical point, this step-size distribution is a power law at this point, with an exponent larger than two, and which the authors conjecture to be equal to 3- beta / nu perpendicular to. beta is the order parameter exponent, and nu perpendicular to is the spatial correlation length exponent. In a finite gradient there is a second exponent governing the upper cut-off in the step-size distribution. This exponent is equal to nu perpendicular to(1+ nu perpendicular to). These two exponents govern the convergence of the position of the front towards the critical value of the control parameter. The authors exemplify their discussion with directed site percolation, and with two versions of a model originally designed to study catalytic reactions.