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

Joe Kiskis·Rajamani Narayanan·Pavlos Vranas
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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 $ν$ describing the vanishing of the physical mass at the critical point is equal to $ν_θ/ d_w$. $d_w$ is the Hausdorff dimension of the walk. $ν_θ$ 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 $ν_θ=φ$, where $φ$ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is $φ/ν$ for O(N) models.

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Joe Kiskis, Rajamani Narayanan, Pavlos Vranas. 1993-03-09. The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory. https://doi.org/10.1007/bf01054349

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