  Inspired by Kubo-Anderson Markov processes, we introduce a new class of
transfer matrices whose largest eigenvalue is determined by a simple explicit
algebraic equation. Applications include the free energy calculation for
various equilibrium systems and a general criterion for perfect harmonicity,
i.e., a free energy that is exactly quadratic in the external field. As an
illustration, we construct a "perfect spring", namely a polymer with
non-Gaussian, exponentially distributed sub-units which nevertheless remains
harmonic until it is fully stretched. This surprising discovery is confirmed by
Monte Carlo and Langevin simulations.
