On the spectrum of random walks on complete finite $d$-ary trees

TL;DR

This paper determines the complete spectrum and eigenbasis of the simple random walk on finite complete d-ary trees, providing insights into their spectral properties and applications to related processes.

Contribution

It uniquely computes the full spectrum and eigenbasis of the transition matrix for random walks on finite complete d-ary trees, a novel spectral analysis.

Findings

Full spectrum of the transition matrix determined

Eigenbasis for the transition matrix constructed

Lower bound for the interchange process established

Abstract

In the present paper, we determine the full spectrum of the simple random walk on finite, complete $d$-ary trees. We also find an eigenbasis for the transition matrix. As an application, we apply our results to get a lower bound for the interchange process on complete, finite d-ary trees, which we conjecture to be sharp.