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We study the random walk representation of the two-point function instatistical mechanics models near the critical point. Using standard scalingarguments we show that the critical exponent $\nu$ describing the vanishing ofthe physical mass at the critical point is equal to $\nu_\theta/ d_w$. $d_w$ isthe Hausdorff dimension of the walk. $\nu_\theta$ is the exponent describingthe 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 impliesthat the Hausdorff dimension of the walk is $\varphi/\nu$ for O(N) models.Comment: 11 pages (plain TeX

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