a rXiv:h ep-th/93 3 55v1 9 M ar 1993UCD-93-06RU-93-09FSU-SCRI-93-37The Hausdorffdimension of random walks and the correlation length critical exponent in Euclidean field theory Joe Kiskis 1,Rajamani Narayanan 2,Pavlos Vranas 3Abstract 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 νdescribing the vanishing of the physical mass at the critical point is equal to νθ/d w .d w is the Hausdorffdimension of the walk.νθis 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 νθ=ϕ,where ϕis the crossover exponent known in thecontext of field theory.This implies that the Hausdorffdimension of the walk is ϕ/νfor O(N)models.KEY WORDS:Random walks in field theory;Hausdorffdimension of random walks;correlation length exponent.
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