A lower bound on the box-counting dimension of crossings in fractal percolation
Following Mandelbrot, the fractal percolation process is introduced as an infinitely repeated subdivision of a square into \(M^2\) subsquares with the retention probability \(p\). The percolating path \(\Gamma \) joining the opposite edges of the original square exists with a non-zero probability if \(p\geq p_c\) and its lower \(\underline{\dim}_B(\Gamma)\) and upper \(\overline{\dim}_B(\Gamma)\) box dimensions are introduced in a usual way. The main result of the paper is the proof of a theorem giving an almost sure lower bound of \(\underline{\dim}_B(\Gamma)\) in the form \(1+v(M)z(z/|\log z|)^3, \;z=1-p\). The theorem is a considerable extension of a result obtained by \textit{L. Chayes} [ibid. 61, No.~1, 25-43 (1996; Zbl 0849.60092)], who proved the non-rectifiability of \(\Gamma \) and the existence of the lower bound of \(\underline{\dim}_B(\Gamma)\) in the form \(1+\zeta , \^^M\zeta >0,\) without specifying the dependence of \(\xi \) on \(M\) and \(p\) explicitly. Another theorem is stated giving an almost sure upper bound of \(\overline{\dim}_B(\Gamma)\) in the immediate vicinity of \(p=1.\) The detailed proof, given in the author's Ph. D. Thesis [University of St. Andrews, 1997], is only sketched here. Even when the both bounds are nontrivial, they are far from being tight.
scientific article; zbMATH DE number 1364456