Universal features of the shapes of percolation clusters and lattice animals

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Abstract
The shapes of percolation clusters and lattice animals are investigated. The universal quantities Delta d and Sd, which were introduced to measure the average asymmetry and degree of prolate- or oblateness, respectively, of long-chain polymers, are here computed in an in expansion for percolation clusters and for lattice animals. Delta d is computed to O( in ), while Sd is computed to O(1). The clusters are shown to be on average anisotropic and prolate, but less so than polymers. Percolation clusters and lattice animals have identical shapes above eight dimensions. Below d=8 animals are slightly more anisotropic than percolation clusters.

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