| Summary: | Percolation theory is usually applied to lattices with a uniform probability p that a site is occupied or that a bond is closed. The more general case, where p is a function of the position x , has received less attention. Previous studies with long-range spatial variations in p ( x ) have only investigated cases where p has a finite, non-zero gradient at the critical point p _c . Here we extend the theory to two-dimensional cases in which the gradient can change from zero to infinity. We present scaling laws for the width and length of the hull (i.e. the boundary of the spanning cluster). We show that the scaling exponents for the width and the length depend on the shape of p ( x ), but they always have a constant ratio 4/3 so that the hull's fractal dimension D = 7/4 is invariant. On this basis, we derive and verify numerically an asymptotic expression for the probability h ( x ) that a site at a given distance x from p _c is on the hull.
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