Two-dimensional percolation with multiple seeds

TL;DR

This paper investigates how multiple seed points affect percolation in a 2D model, revealing that the percolation threshold rises with seed concentration while critical exponents remain unchanged, and identifying unique finite-size effects at low seed levels.

Contribution

It introduces a study of non-uniform percolation with multiple seeds, showing invariant critical exponents and the persistence of scaling laws across seed concentrations.

Findings

Percolation threshold increases with seed concentration

Critical exponents remain invariant across seed densities

Finite-size scaling anomalies occur at low seed concentrations

Abstract

We study non-uniform percolation in a two-dimensional cluster growth model with multiple seeds. With increasing concentration of seeds, the percolation threshold is found to increase monotonically, while the exponents for correlation length, order parameter, and average cluster size, keep invariant. The scaling law for an infinite square lattice keeps working for any nonzero concentration of seeds. Abnormal finite-size scaling behaviours happen at low concentration of seeds.