Spectrum of an oscillator making a random walk in frequency

The spectrum of a spectroscopically active particle which makes a random walk on a finite one-dimensional lattice, upon each of whose n sites the particle radiates or absorbs radiation with a different frequency, is discussed. The correlation function of the radiating dipole is found to decay asymptotically as a Gaussian times $t^{\mathrm{-}\left(n\mathrm{-}1\right)/2}$ in the case when the site frequencies are independent Gaussian random variables. Numerical simulations of the random walk confirm the asymptotic result. The difference between these results for a finite system and previous results for an infinite system are discussed.