Spectral random walk of a single molecule

We investigate the fluctuations in the transition energy of a single molecule embedded in a solid. We propose a model based on spatially distributed two-level systems with internal dynamics which induce the fluctuations due to long-range interactions with the single molecule. We show that the transition energies follow a spectral random walk which can be analyzed in terms of random walk theories. The corresponding spectral propagator is derived and its relationship to stable distributions is established. For the particular case of dipole interactions in a three-dimensional system the Cauchy distribution is recovered.