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 $ν$ 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.