# The length of self-avoiding walks on the complete graph

## TL;DR

This paper analyzes the asymptotic behavior of self-avoiding walks on complete graphs, deriving mean, variance, and distributional limits of walk length as the graph size grows, especially near critical points.

## Contribution

It provides the first detailed asymptotic analysis of self-avoiding walk lengths on complete graphs, including central limit theorems and effects of fugacity convergence.

## Key findings

- Asymptotic mean and variance of walk length derived
- Central limit theorems established in various regimes
- Effects of fugacity convergence on walk length analyzed

## Abstract

We study the variable-length ensemble of self-avoiding walks on the complete graph. We obtain the leading order asymptotics of the mean and variance of the walk length, as the number of vertices goes to infinity. Central limit theorems for the walk length are also established, in various regimes of fugacity. Particular attention is given to sequences of fugacities that converge to the critical point, and the effect of the rate of convergence of these fugacity sequences on the limiting walk length is studied in detail. Physically, this corresponds to studying the asymptotic walk length on a general class of pseudocritical points.