Random Walks and the Correlation Length Critical Exponent in Scalar Quantum Field Theory

TL;DR

This paper explores the relationship between random walk representations and the correlation length critical exponent in scalar quantum field theory, revealing how walk properties relate to critical behavior near phase transitions.

Contribution

It establishes a connection between the properties of random walks and the critical exponents in scalar quantum field theory, providing a new understanding of the walk dimension and energy-length relation.

Findings

Walk dimension d_w = φ / ν.

Singular relation θ ∼ t^φ between energy per length and t.

Connection between walk properties and two-point function behavior.

Abstract

The distance scale for a quantum field theory is the correlation length $\xi$, which diverges with exponent $\nu$ as the bare mass approaches a critical value. If $t=m^{2}-m_{c}^{2}$, then $\xi=m_{P}^{-1} \sim t^{-\nu}$ as $t \to 0$. The two-point function of a scalar field has a random walk representation. The walk takes place in a background of fluctuations (closed walks) of the field itself. We describe the connection between properties of the walk and of the two-point function. Using the known behavior of the two point function, we deduce that the dimension of the walk is $d_{w}=\phi / \nu$ and that there is a singular relation between $t$ and the energy per unit length of the walk $\theta \sim t^{\phi}$ that is due to the singular behavior of the background at $t=0$. ($\phi$ is a computable crossover exponent.)