We consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior. Bootstrap percolation is an example of Cellular Automata defined on the d-dimensional lattice {1, 2, ..., L} d in which each site can be empty or occupied by a single particle; in the starting configuration each site is occupied with probability p, occupied sites remain occupied for ever, while empty sites are occupied by a particle if at least ℓ among their 2d nearest neighbor sites are occupied. When d is fixed, the most interesting case is the one ℓ = d: this is a sort of threshold, in the sense that the critical probability p c for the dynamics on the infinite lattice Z d switches from zero to one when this limit is crossed. Finite size effects in the three-dimensional case are already known in the cases ℓ ≤ 2: in this paper we discuss the case ℓ = 3 and we show that the finite size scaling function for this problem is of the form f (L) = const/ ln ln L. We prove a conjecture proposed by A.C.D. van Enter.

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We compute the expansion of the surface tension of the 3D random cluster model for q ≥ 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n goes to ∞. This same shape determines the Wulff crystal to order o(ε) in the 3D Ising model (and more generally in the 3D random cluster model for q ≥ 1) at temperature ε.

The set of the three dimensional polyominoes of minimal area and of volume $n$ contains a polyomino which is the union of a quasicube $j\times (j+\delta)\times (j+\theta)$, $\delta,\theta\in\{0,1\}$, a quasisquare $l\times (l+\epsilon)$, $\epsilon\in\{0,1\}$, and a bar $k$. This shape is naturally associated to the unique decomposition of $n=j(j+\delta)(j+\theta)+l(l+\epsilon)+k$ as the sum of a maximal quasicube, a maximal quasisquare and a bar. For $n$ a quasicube plus a quasisquare, or a quasicube minus one, the minimal polyominoes are reduced to these shapes. The minimal area is explicitly computed and yields a discrete isoperimetric inequality. These variational problems are the key for finding the path of escape from the metastable state for the three dimensional Ising model at very low temperatures. The results and proofs are illustrated by a lot of pictures.