Math @ Duke

Publications [#243445] of Richard T. Durrett
Papers Published
 Chayes, JT; Chayes, L; Durrett, R, Connectivity properties of Mandelbrot's percolation process,
Probability Theory and Related Fields, vol. 77 no. 3
(1988),
pp. 307324, ISSN 01788051 [doi]
(last updated on 2017/12/17)
Abstract: 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 pc∈(0,1) so that if p<pc then the set is "duslike" i.e., the largest connected component is a point, whereas if p≧pc (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. © 1988 SpringerVerlag.


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