Euler Walk on a Cayley Tree

TL;DR

This paper analyzes the behavior of Euler walks on Cayley trees, revealing two distinct phases with different growth patterns and characterizing the critical transition between them.

Contribution

It introduces a detailed analysis of the phase transition and growth dynamics of Euler walks on Cayley trees, including return behavior and critical phenomena.

Findings

Identification of condensed and low-density phases

Characterization of growth patterns in each phase

Analysis of critical behavior at phase transition

Abstract

We show that the Euler walk on a Cayley tree exhibits two regimes (dynamic phases): a condensed phase and a low-density phase. In the condensed phase the self-organized area grows as a compact domain. In the low-density phase the proportion of self-organized (visited) nodes decreases rapidly from one generation of the tree to the next. We describe in detail returns of the Euler walk to the root and growth of the self-organized domain in the condensed phase. We also investigate the critical behaviour of the Euler walk at the point separating the two regimes.