Electronic Communications in Probability

A spectral decomposition for a simple mutation model

Martin Möhle

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We consider a population of $N$ individuals. Each individual has a type belonging to some at most countable type space $K$. At each time step each individual of type $k\in K$ mutates to type $l\in K$ independently of the other individuals with probability $m_{k,l}$. It is shown that the associated empirical measure process is Markovian. For the two-type case $K=\{0,1\}$ we derive an explicit spectral decomposition for the transition matrix $P$ of the Markov chain $Y=(Y_n)_{n\ge 0}$, where $Y_n$ denotes the number of individuals of type $1$ at time $n$. The result in particular shows that $P$ has eigenvalues $(1-m_{0,1}-m_{1,0})^i$, $i\in \{0,\ldots ,N\}$. Applications to mean first passage times are provided.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 15, 14 pp.

Received: 11 August 2018
Accepted: 26 February 2019
First available in Project Euclid: 21 March 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 92D10: Genetics {For genetic algebras, see 17D92}

eigenvalues eigenvectors empirical measure process finite Markov chain first passage time mixing time mutation model potential theory product chain random walk on the hypercube spectral analysis

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Möhle, Martin. A spectral decomposition for a simple mutation model. Electron. Commun. Probab. 24 (2019), paper no. 15, 14 pp. doi:10.1214/19-ECP222. https://projecteuclid.org/euclid.ecp/1553133704

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