Exact results at the two-dimensional percolation point

Exact Results at the 2 - D Percolation Point

We derive exact expressions for the excess number of clusters b and the excess cumulants b n of a related quantity at the 2-D percolation point. High-accuracy computer simulations are in accord with our predictions. b is a finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula for the number of clusters per site n c in the infinite system limit; the b n correct bulk cum...

Exact Determination of the Two-Point Cluster Function for One-Dimensional Continuum Percolation

The two-point cluster function C2(rl, r2) provides a measure of clustering in continuum models of disordered many-particle systems and thus is a useful signature of the microstructure. For a two-phase disordered medium, C2(rt, r2) is defined to be the probability of finding two points at positions r t and r_, in the same cluster o f one o f the phases. An exact ana/ytical expression is found fo...

Two-Dimensional Critical Percolation

We use SLE6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved. In the m...

Two Results about Fixed Point of Multifunctions

Exact bond percolation thresholds in two dimensions

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose perco...