A spectral decomposition for a simple mutation model. (English) Zbl 1412.60106

Möhle, Martin

Electron. Commun. Probab. 24, Paper No. 15, 14 p. (2019).

Summary: 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.

MSC:

60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
15A18	Eigenvalues, singular values, and eigenvectors
60J45	Probabilistic potential theory
92D10	Genetics and epigenetics

Keywords:

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

References:

[1]	Berger, M. and Cerf, R.: A basic model of mutations, (2018), arXiv:1806.01212 · Zbl 1455.60094
[2]	Burke, C. J. and Rosenblatt, M.: A Markovian function of a Markov chain. Ann. Math. Stat.29, (1958), 1112-1122. · Zbl 0100.34402 · doi:10.1214/aoms/1177706444
[3]	Cannings, C.: The latent roots of certain Markov chains arising in genetics: a new approach, I. Haploid models. Adv. Appl. Probab.6, (1974), 260-290. · Zbl 0284.60064 · doi:10.1017/S0001867800045365
[4]	Cannings, C.: The latent roots of certain Markov chains arising in genetics: a new approach, II. Further haploid models. Adv. Appl. Probab.7, (1975), 264-282. · Zbl 0339.60067 · doi:10.2307/1426077
[5]	Ehrenfest, P. and Ehrenfest, T.: Über zwei bekannte Einwände gegen das Boltzmannsche \(H\)-Theorem. Phys. Z.8, (1907), 311-314. · JFM 38.0931.01
[6]	Gladstien, K.: The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM J. Appl. Math.34, (1978), 630-642. · Zbl 0386.92004 · doi:10.1137/0134050
[7]	Grinstead, C. M. and Snell J. L.: Introduction to Probability. Second Revised Version. American Mathematical Society, 1997. · Zbl 0914.60004
[8]	Kac, M.: Random walk and the theory of Brownian motion. Amer. Math. Monthly54, (1947), 369-391. · Zbl 0031.22604 · doi:10.1080/00029890.1947.11990189
[9]	Levin, D. and Peres, Y.: Markov Chains and Mixing Times. American Mathematical Society, Providence, 2017. · Zbl 1390.60001
[10]	Nestoridi, E.: A non-local random walk on the hypercube. Adv. Appl. Probab.49, (2017), 1288-1299. · Zbl 1423.60111 · doi:10.1017/apr.2017.42
[11]	Norris, J. R.: Markov Chains. Cambridge Studies in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 1998. · Zbl 0938.60058
[12]	Scoppola, B.: Exact solution for a class of random walks on the hypercube. J. Stat. Phys.143, (2011), 413-419. · Zbl 1219.82097 · doi:10.1007/s10955-011-0194-y