Percolation fractal exponents without scaling

Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called ”hull” or external perimeter of the incipient percolation cluster keeps its value in describing the width and length of gradient percolation frontiers whatever the gradient value. Its origin is then not to be found in the thermodynamic limit. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law from size 1 to infinity which describes the gradient percolation frontier, this law becoming a scaling law in the large system limit. Spreading of objects in space with a gradient of probability is most common. From chemical composition gradients to the distribution of plants which depends of their solar exposure, probability gradients exist in many inhomogeneous systems. In fact inhomogeneity is a rule in nature whereas most of the systems that physicists are studying are homogeneous as they are thought to be more simple to understand. In particular phase transitions or critical phenomena are studied in that framework, the simplest being percolation transition[1]. In this work, the opposite situation, a strongly inhomogeneous system is studied. We report the discovery that the validity of some percolation exponents can be extended to situations which are very far from the large homogeneous system limit. This is found in the frame of gradient percolation, a situation first introduced in the study of diffusion front[2, 3]. Surprisingly these exponents, up to now believed to pertain to large systems, are also verified in a limit that could be called the small system limit (S.S.L.) where some of the gradient percolation properties can be computed analytically. This letter presents a report concerning the 2D square lattice. The same results and discussion have been obtained for the triangular lattice and will be published elsewhere, together with the details of the analytical calculations for both lattices. The gradient percolation (G.P.) situation is depicted in FIG. 1. The figure gives an example of a random distribution of points on a lattice with a linear gradient of concentration in the vertical direction. It is a 2D square lattice of size Lg × L, where each point (x, y) is occupied with probability p(x) = 1 − x/Lg (x being the vertical direction in the figure). In gradient percolation there is always an infinite cluster of occupied sites as there is a region