Functional integral representations for self-avoiding walk

TL;DR

This paper surveys functional integral representations for various self-avoiding walk models, introduces a new representation for strictly self-avoiding walks, and discusses applications in renormalization group analysis.

Contribution

It provides a unified treatment of integral representations for self-avoiding walks, including a novel representation for strictly self-avoiding walks and an introduction to fermionic integrals.

Findings

New integral representation for strictly self-avoiding walk.

Applications of integral representations in renormalization group analysis.

Introduction to fermionic integrals using Grassmann variables.

Abstract

We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms.