Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process
نویسندگان
چکیده
منابع مشابه
Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot’s Fractal Percolation Process
We study Mandelbrot’s percolation process in dimension d ≥ 2. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1]d in Nd subcubes, and independently retaining or discarding each subcube with probability p or 1− p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolatio...
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The geometrical explanation of universality in terms of fixed points of renormalization-group transformations has met with enormous success, but its lack of precision continues to present a challenge to the mathematician, even for relatively simple models, such as percolation in two dimensions. To begin to reflect on the problem, one can assume that the crossing probabilities (cf. [5]) at the c...
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چکیده: فرض کنید که تابعی از اپسیلون یک مجموع نامتناهی از احتمالات موزون مربوط به مجموع های جزئی براساس یک دنباله از متغیرهای تصادفی مستقل و همتوزیع باشد، و همچنین فرض کنید توابعی مانند g و h وجود دارند که هرگاه امید ریاضی توان دوم x متناهی و امیدریاضی x صفر باشد، در این صورت می توان حد حاصلضرب این توابع را بصورت تابعی از امید ریاضی توان دوم x نوشت. حالت عکس نیز برقرار است. همچنین ما با استفاده...
15 صفحه اولUniversal crossing probabilities and incipient spanning clusters in directed percolation
Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. In a dynamical interpretation, the crossing probability is the probability that, on a system with size L, an epidemic spreading without immunization remains active at time t. Since the sys...
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ژورنال
عنوان ژورنال: Electronic Journal of Probability
سال: 2008
ISSN: 1083-6489
DOI: 10.1214/ejp.v13-511