Renormalisation group approach to Delaunay percolation networks with topological disorder

Abstract

A renormalisation group approach is developed for Delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space. Considering various renormalisation transformations for two- and three-dimensional Delaunay lattices, the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent. The fixed points for the two-dimensional bond lattice and the three-dimensional site lattice are 0.3229 and 0.1443 respectively, which are in excellent agreement with results of Monte Carlo simulations. The fixed point for the two-dimensional site lattice gives the value 1/2 for the critical percolation probability which is equal to the known result for a fully triangulated lattice.