Suppose that n identical particles evolve according to the same marginal
Markov chain. In this setting we study chains such as the Ehrenfest chain
that move a prescribed number of randomly chosen particles at each epoch.
The product chain constructed by this device inherits its eigenstructure
from the marginal chain. There is a further chain derived from the product
chain called the composition chain that ignores particle labels and tracks
the numbers of particles in the various states. The composition chain in
turn inherits its eigenstructure and various properties such as
reversibility from the product chain. The equilibrium distribution of the
composition chain is multinomial. The current paper proves these facts in
the well-known framework of state lumping and identifies the column
eigenvectors of the composition chain with the multivariate Krawtchouk
polynomials of Griffiths. The advantages of knowing the full spectral
decomposition of the composition chain include (a) detailed estimates of
the rate of convergence to equilibrium, (b) construction of martingales
that allow calculation of the moments of the particle counts, and
(c) explicit expressions for mean coalescence times in multi-person random
walks. These possibilities are illustrated by applications to Ehrenfest
chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA
evolution, a light bulb chain, and random walks on some specific graphs.
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