The crossover from 2D to 3D percolation: Theory and numerical simulations

We describe here the crossover between 2D and 3D percolation, which we do on cubic and square lattices. As in all problems of critical phenomena, the quantities of interest can be expressed as power laws of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert p-p_{\rm c}(h)\vert$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{\rm c}(h)$\end{document} and h are the percolation threshold and the thickness of the film, respectively. When these quantities are considered on the scale of the thickness h of the films, the corresponding numerical prefactors are of order one. However, for many problems, the scale of interest is the elementary one. The corresponding expressions contain then prefactors in power of h which we calculate. For instance, we show that the mass distribution n(m) of the clusters is given by a master function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h^{-D + 1/\sigma_{2}\nu_{3}}\vert p-p_{\rm c}(h)\vert^{1/\sigma_{2}} m$\end{document}, where h is the thickness of the film and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D,\nu_3,\sigma_2$\end{document} are tabulated 2D and 3D critical exponents. We consider also the size R2(m) of the clusters as a function of their mass m, for which we provide both scaling laws and numerical data. Therefore, any property corresponding to a given moment of mass and size can be obtained from our results. These results might be useful for describing transport properties, such as electric conductivity, or the mechanical properties of thin films made of disordered materials.