The fractal volume of the two-dimensional invasion percolation cluster

Abstract

We consider both invasion percolation and standard Bernoulli bond percolation on theZ ² lattice. Denote byV andC the invasion cluster and the occupied cluster of the origin, respectively. Let$\mathcal{V}_n = \mathcal{V} \cap [ - n, n]^2$, and

Let ε>0 be given. Here we show that, with a probability tending to 1,

$$n^{2 - \varepsilon } \pi _n \leqq |\mathcal{V}_n | \leqq n^{2 + \varepsilon } \pi _n .$$

Assuming the existence of an exponent 1/ρ for π _n it can be seen that with probability tending to one

$$n^{2 - 1/\rho - \varepsilon } \leqq |\mathcal{V}_n | \leqq n^{2 - 1/\rho + \varepsilon } .$$

Moreover, by den Nijs' and Nienhuis et al's computations,

$$n^{1.8958389583... - \varepsilon } = n^{1 + \tfrac{{43}}{{48}} - \varepsilon } \leqq |\mathcal{V}_n | \leqq n^{1 + \tfrac{{43}}{{48}} - \varepsilon } = n^{1.8958389583... + \varepsilon } $$

with a probability tending to one. The result matches Wilkinson and Willemsen's numerical computation$\mathcal{V}_n \sim n^{1.89}$. The method allows us also to show the same argument for any planar graph. Therefore, any two planar invasion clusters have the same fractal dimension 2−1/ρ if one believes “universality”.

Furthermore, the escape time of the invasion cluster is considered in this paper. More precisely, denote byh _n the first time that the invasion cluster escapes from [−n,n]². We here can show that with a probability tending to one

$$n^{2 - \varepsilon } \pi _n \leqq h_n \leqq n^{2 + \varepsilon } \pi _n .$$

Finally, invasion percolation with trapping is considered in this paper. Denote by

$$\mathcal{R}_n = \{ the number of bonds trapped by the invasion cluster before time n\} $$

Here we show that with a probability tending to one

$$n^{2/(2 - \alpha _n ) - \varepsilon } \leqq |\mathcal{R}_n | \leqq n^{2/(2 - \alpha _n ) + \varepsilon } ,$$

where$\alpha _n = - \frac{{\log \pi _n }}{{\log _n }}$. By assuming the existence of ρ and$\rho = \tfrac{{48}}{5}$ again, we can show that

$$n^{1.054945054945... + \varepsilon } \leqq |\mathcal{R}_n | \leqq n^{2\rho /(2\rho - 1) + \varepsilon } = n^{1.054945054945... + \varepsilon } $$

with a probability tending to one.