Stationary Distribution and Eigenvalues for a de Bruijn Process
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.
KeywordsCorrelation Function Markov Chain Stationary Distribution Characteristic Polynomial Product Measure
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We thank the referee for a careful reading of the manuscript. The first author (A.A.) would like to acknowledge hospitality and support from the Tata Institute of Fundamental Research, Mumbai, India where part of this work was done, and thank T. Amdeberhan for discussions.
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