Abstract:
Percolation models in which the centers of defects are distributed randomly in space
in accordance with Poisson's law and the shape of each defect is also random are
considered. Methods of obtaining rigorous estimates of the critical densities are
described. It is shown that the number of infinite clusters can take only three values:
0, 1, or $\infty$. Models in which the defects have an elongated shape and a random
orientation are investigated in detail. In the two-dimensional case, it is shown that
the critical volume concentration of the defects is proportional to $a/l$, where $l$ and
$a$ are, respectively, the major and minor axes of the defect; the mean number of
(direct) bonds per defect when percolation occurs is bounded.
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