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

TL;DR

This paper links the Hausdorff dimension of random walks to critical exponents in Euclidean field theory, providing a geometric interpretation of critical phenomena in statistical mechanics models.

Contribution

It establishes a relationship between the Hausdorff dimension of walks and critical exponents, specifically connecting $ u$, $ u_ heta$, and the crossover exponent $$, in O(N) models.

Findings

The critical exponent $ u$ equals $ u_ heta/d_w$.

For O(N) models, $ u_ heta$ equals the crossover exponent $$.

The Hausdorff dimension of the walk is $/ u$ in O(N) models.

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 $\nu$ 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.