HAUSDORFF DIMENSION OF CONTINUOUS POLYAKOV’S RANDOM SURFACES

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Abstract
In this paper we compute exactly, using the scaling properties of the Liouville theory, the Hausdorff dimension of the continuous random surfaces of Polyakov for D≤1. We find that for D<1, the mean square size of the surface grows as a logarithm of the area of the surface as well as the area of the surface raised to a power, the power being minus the string susceptibility. For D=1 the behavior changes, as expected, because the model undergoes a phase transition. In that case we find that the mean square size of the surface behaves as a combination of terms that grow as a logarithm of the area as well as its square, in qualitative agreement with the results of numerical experiments in discrete models.

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