The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory

Abstract

We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments we show that the critical exponent $\nu$ describing the vanishing of the physical mass at the critical point is equal to ${\nu }_{\theta }/d_w$. $d_w$ is the Hausdorff dimension of the walk. ${\nu }_{\theta }$ is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case of O(N) models, we show that ${\nu }_{\theta }=\phi$, where $\phi$ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is $\phi /\nu$ for O(N) models.

11 pages (plain TeX)

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