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

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

1993-03-09

arxiv.org/abs/hep-th/9303055v1

Physics

High Energy Physics

High Energy Physics - Theory

11 pages (plain TeX)

Scientific paper

10.1007/BF01054349

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 $\nu_\theta=\varphi$, where $\varphi$ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is $\varphi/\nu$ for O(N) models.

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