On Mixing Behavior of a Family of Random Walks Determined by a Linear Recurrence

TL;DR

This paper analyzes the mixing times of a family of integer-based random walks defined by linear recurrence sequences, using Fourier analysis to derive explicit eigenvalue formulas and bounds.

Contribution

It introduces a method to compute eigenvalues of transition matrices for these walks, enabling precise mixing time bounds based on the recurrence relation.

Findings

Explicit eigenvalue formulas derived via Fourier analysis

Bounding mixing times based on the recurrence sequence

Applicable to a broad class of linear recurrence-defined walks

Abstract

We study random walks on the integers mod $G_n$ that are determined by an integer sequence $\{ G_n \}_{n \geq 1}$ generated by a linear recurrence relation. Fourier analysis provides explicit formulas to compute the eigenvalues of the transition matrices and we use this to bound the mixing time of the random walks.