On the critical value function in the divide and color model
Journal article, 2013

The divide and color model on a graph G arises by first deleting each edge of G with probability 1-p independently of each other, then coloring the resulting connected components (i.e., every vertex in the component) black or white with respective probabilities r and 1-r, independently for different components. Viewing it as a (dependent) site percolation model, one can denote the critical point r^G_c(p). In this paper, we mainly study the continuity properties of the function r^G_c, which is an instance of the question of locality for percolation. Our main result is the fact that in the case G=Z^2, r^G_c is continuous on the interval [0,1/2); we also prove continuity at p=0 for the more general class of graphs with bounded degree. We then investigate the sharpness of the bounded degree condition and the monotonicity of r^G_c(p) as a function of p.

Percolation

critical value

divide and color model

stochastic domination

locality

Author

András Bálint

Chalmers, Applied Mechanics, Vehicle Safety

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

SAFER, The Vehicle and Traffic Safety Centre

Vincent Beffara

Vincent Tassion

Alea

1980-0436 (ISSN)

Vol. 10 2 653-666

Roots

Basic sciences

Subject Categories

Other Physics Topics

Probability Theory and Statistics

More information

Created

10/7/2017