We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset {\mathbb{R}}^k$ ($k\ge 2$) obtained by using some percolation process in ${[0,1]}^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on K. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing ${\dim }_{H}K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on K, and show that the supremum is uniquely attained. The value of ${\dim }_{H}K$ is also expressed in terms of the weighted pressure function of some deterministic potential. As a by-product, when $k=2$, we give an alternative approach to the Hausdorff dimension of K, which was first obtained by Gatzouras and Lalley [27]. The value of the box counting dimension of K and its equality with ${\dim }_{H}K$ are also studied. We also obtain a variational formula for the Hausdorff dimensions of some orthogonal projections of K, and for statistically self-affine measures supported on K, we establish a dimension conservation property through these projections.

我们计算任何随机统计自仿射Sierpinski海绵的Hausdorff尺寸 $ķ\subset {\mathbb{[R}}^{ķ}$ （$ķ\ge \mathrm{2个}$）是通过在 ${[0，\mathrm{1个}]}^{ķ}$。为此，我们首先展示在K上支持的统计自仿射度量的Hausdorff维数的Ledrappier-Young型公式。与确定性或随机动态版本相比，此公式提供了一个新功能。然后，我们建立了一个变分原理表示${\mathrm{暗淡}}_Hķ$作为支持K的统计自仿射度量的Hausdorff维数的极值，并表明极值是唯一获得的。的价值${\mathrm{暗淡}}_Hķ$用一些确定性势的加权压力函数表示。作为副产品，当$ķ=\mathrm{2个}$，我们为K的Hausdorff维数提供了另一种方法，该方法最初是由Gatzouras和Lalley [27]获得的。K的盒数维数的值及其与${\mathrm{暗淡}}_Hķ$也进行了研究。我们也得到了一些正交投影的豪斯多夫尺寸的变分公式ķ，并支持统计自仿射措施ķ，我们通过建立这些预测维度的保护特性。