# Self-avoiding walk on the complete graph

## TL;DR

This paper analyzes the phase transition of self-avoiding walks on the complete graph, providing exact susceptibility formulas and limiting distributions across different regimes, serving as a prototype for more complex structures.

## Contribution

It offers the first detailed finite-size scaling analysis of self-avoiding walks on the complete graph, including exact susceptibility and distribution results.

## Key findings

- Susceptibility expressed via incomplete gamma function
- Complete description of finite-size scaling behavior
- Limiting distributions in various regimes

## Abstract

There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a high-dimensional torus.