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{"work":{"id":69177704,"created_at":"2022-01-22T09:14:24.584-08:00","from_world_paper_id":193032289,"updated_at":"2024-11-25T05:59:21.739-08:00","_data":{"publisher":"Institute of Mathematical Statistics","grobid_abstract":"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 N d 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 percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)-dimensional \"sheets\" for all d ≥ 3. For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that p c (N, 2) converges, as N → ∞, to the critical density p c of site percolation on the square lattice. Assuming the existence of","publication_date":"2008,,","publication_name":"Electronic Journal of Probability","grobid_abstract_attachment_id":"79372642"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [52848710]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div 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In these objects the multifractality comes directly from the geometric tiling. We identify some differences between percolation in the proposed multifractals and in a regular lattice. There are basically two sources of these differences. The first is related to the coordination number, which changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. We observe that the percolation threshold for the multifractal is lower than that for the square lattice. We compute the fractal dimension of the percolating cluster and the critical exponent β. 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