Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process
| dc.creator | Broman, Erik I. | |
| dc.creator | Camia, Federico | |
| dc.date | 2007-12-20 | |
| dc.date | 2008-02-22 | |
| dc.date.accessioned | 2026-07-07T09:22:13Z | |
| dc.date.available | 2026-07-07T09:22:13Z | |
| dc.description | We study Mandelbrot's percolation process in dimension $d \geq 2$. 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 $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, 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 $N \to \infty$. 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 $N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-percolation. In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N \to \infty$, to the critical density $p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $ν$ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $p_c(N,2)-p_c=(\frac{1}{N})^{1/ν+o(1)}$ as $N \to \infty$, showing an interesting relation with near-critical percolation. | |
| dc.description | 24 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/0712.3422 | |
| dc.identifier | http://arxiv.org/abs/0712.3422 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155300 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | 60K35, 60D05, 28A80, 82B43 | |
| dc.title | Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process | |
| dc.type | text |