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 exponentv describing the vanishing of the physical mass at the critical point is equal tov _Θ /d_w, whered _w is the Hausdorff dimension of the walk, andv _Θ is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case ofO(N) models, we show thatv ₀=ϕ, whereϕ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk isϕ/v forO(N) models.