The full spectrum of random walks on complete finite d-ary trees

Evita Nestoridi,Oanh Nguyen

doi:10.1214/21-ejp608

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.

Highlights

Finding the spectrum of a transition matrix is a very popular subject in graph theory and Markov chain theory
There are only a few techniques known to describe the exact spectrum of a Markov chain, and they usually work under very specific conditions, such as when the Markov chain is a random walk on a finite group, generated only by a conjugacy class [12]
We present the full spectrum of the simple random walk on complete, finite d-ary trees and a corresponding eigenbasis, and we use this information to produce a lower bound for the interchange process on the trees, which we conjecture is sharp

Summary

Introduction

Finding the spectrum of a transition matrix is a very popular subject in graph theory and Markov chain theory. The second largest eigenvalue can be derived by this new process, while the eigenvectors are of the form presented in (1.5) The reason why this is the right process to study is hidden in the mixing time of the random walk on Th. A coupling argument roughly says that we have to wait until Xt reaches the root ρ. A configuration of the deck corresponds to an element of the symmetric group, which we denote by Sn where we recall that n is the number of vertices of Th. Let g ∈ Sn. Let P be the transition matrix of the interchange process on the complete, finite d-ary tree Th and let Pitd(g) be the probability that we are at g after t steps, given that we start at the identity. We suggest trying to find the spectrum or just the exact value of the spectral gap for the simple random on finite Galton-Watson trees or for the frog model as presented in [16]

The spectrum of Qh

Completely symmetric eigenvectors

Pseudo anti-symmetric eigenvectors