The cross-over from 2D to 3D percolation and its relationship to glass transition in thin films. Theory and numerical simulations

We consider here the percolation problem in thin films, both in the direction normal to the film and in the direction parallel to the film. We thereby describe here the cross-over between 2D and 3D percolation, which we do on cubic and square lattices. The main relations are derived using scaling and real space renormalisation arguments. They are checked by numerical simulations, which also provide the numerical prefactors. We calculate in particular the correlation length parallel to the film, the average mass and the mass distribution $n(m)$ of the clusters. In particular, we show that the latter is given by a master function of $h^{-D+1/\sigma_{2}\nu_{3}}| p-p_{c}(h)|^{1/\sigma_{2}} m$, where $h$ is the thickness of the film and $D,\nu_3,\sigma_2$ are tabulated 2D and 3D critical exponents. $p_c(h)$ is the percolation threshold of the film which we also calculate. These results are of interest in particular for describing the glass transition in thin polymer films.