Smooth Paths on Three Dimensional Lattice

TL;DR

This paper investigates a class of three-dimensional lattice random walks with spin factors, calculating critical parameters and constructing continuum field theories near critical points, extending concepts from 2D Ising models.

Contribution

It introduces a three-dimensional random walk model with spin, computes critical diffusion constants and exponents, and develops related continuum field theories.

Findings

Critical diffusion constants and exponents are calculated.

Continuum field theories like Klein-Gordon, Dirac, and Chern-Simons are constructed.

Model generalizes 2D Ising random walk to 3D.

Abstract

A particular class of random walks with a spin factor on a three dimensional cubic lattice is studied. This three dimensional random walk model is a simple generalization of random walk for the two dimensional Ising model. All critical diffusion constants and associated critical exponents are calculated. Continuum field theories such as Klein-Gordon, Dirac and massive Chern-Simons theories are constructed near several critical points.