Stationary Distribution and Eigenvalues for a de Bruijn Process

TL;DR

This paper analyzes a de Bruijn process modeled as a continuous-time Markov chain, explicitly determines its steady state and eigenvalues, and explores special cases with product and correlated measures.

Contribution

It provides explicit formulas for the stationary distribution and eigenvalues of the de Bruijn process, including special cases with unique correlation structures.

Findings

Steady state distribution explicitly computed.

Characteristic polynomial decomposes into linear factors.

Two special cases analyzed: product measure and correlated distribution.

Abstract

We define a de Bruijn process with parameters n and L as a certain continuous-time Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as vertices. We determine explicitly its steady state distribution and its characteristic polynomial, which turns out to decompose into linear factors. In addition, we examine the stationary state of two specializations in detail. In the first one, the de Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de Bruin process, the distribution has constant density but nontrivial correlation functions. The two point correlation function is determined using generating function techniques.