What is the difference between a square and a triangle?

TL;DR

This paper explains the attracting edge problem in strongly edge reinforced random walks on squares and even cycles, comparing two techniques and highlighting challenges with triangles.

Contribution

It provides a clear comparison of two approaches to verifying the attracting edge phenomenon in SERRW on squares and cycles, and discusses limitations with triangles.

Findings

Both techniques extend to even cycles.

The attracting edge exists almost surely for squares.

The triangle case presents unique challenges.

Abstract

We offer a reader-friendly introduction to the attracting edge problem (also known as the "triangle conjecture") and its most general current solution of Limic and Tarr\`es (2007). Little original research is reported; rather this article ``zooms in'' to describe the essential characteristics of two different techniques/approaches verifying the almost sure existence of the attracting edge for the strongly edge reinforced random walk (SERRW) on a square. Both arguments extend straightforwardly to the SERRW on even cycles. Finally, we show that the case where the underlying graph is a triangle cannot be studied by a simple modification of either of the two techniques.