, Volume 77, Issue 3, pp 307-324

Connectivity properties of Mandelbrot's percolation process

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Summary

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 self-similar 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 pp 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.

Work supported by the NSF under Grant #DMR-83-14625
Work supported by the DOE under Grant #DE-AC02-83-ER13044
Work supported by the NSF under Grant #DMS-85-05020