The geometry of percolation fronts in two-dimensional lattices with spatially varying densities

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.

💡 Research Summary

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The paper extends classical percolation theory, which traditionally assumes a uniform occupation probability p, to the far more realistic situation where p varies with position x in a two‑dimensional lattice. While earlier works have examined spatial gradients that are finite and non‑zero at the critical threshold p_c, this study systematically treats the full spectrum of possible gradients, ranging from zero (locally flat p(x)) to arbitrarily large (sharp jumps).

The authors begin by representing the spatial dependence of the occupation probability as a power‑law deviation from criticality:
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