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

TL;DR

This paper extends percolation theory to two-dimensional lattices with spatially varying occupation probabilities, deriving scaling laws for hull properties and revealing invariant fractal dimensions despite different gradient conditions.

Contribution

It introduces a generalized framework for percolation with variable p(x), including cases with zero or infinite gradients, and establishes universal scaling relations and hull probability expressions.

Findings

Scaling laws for hull width and length depending on p(x) shape

Invariant fractal dimension D=7/4 for the hull

Numerical verification of asymptotic hull probability expression

Abstract

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.