Connectivity properties of Mandelbrot's percolation process
 J. T. Chayes,
 L. Chayes,
 R. Durrett
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In 1974, Mandelbrot introduced a process in [0, 1]^{2} which he called “canonical curdling” and later used in this book(s) on fractals to generate selfsimilar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivity or “percolation” properties of these sets, proving all of the claims he made in Sect. 23 of the “Fractal Geometry of Nature” and a new one that he did not anticipate: There is a probability p _{c}∈(0,1) so that if p<p _{c} then the set is “duslike” i.e., the largest connected component is a point, whereas if p≧p _{c} (notice the =) opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles. More succinctly put the system has a first order phase transition.
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 Title
 Connectivity properties of Mandelbrot's percolation process
 Journal

Probability Theory and Related Fields
Volume 77, Issue 3 , pp 307324
 Cover Date
 19880301
 DOI
 10.1007/BF00319291
 Print ISSN
 01788051
 Online ISSN
 14322064
 Publisher
 SpringerVerlag
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 Authors

 J. T. Chayes ^{(1)}
 L. Chayes ^{(1)}
 R. Durrett ^{(2)}
 Author Affiliations

 1. Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, White Hall, 148537901, NY, USA
 2. Department of Mathematics, Cornell University Ithaca, White Hall, 148537901, NY, USA