  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.
