  Applied to statistical physics models, the random cost algorithm enforces a
Random Walk (RW) in energy (or possibly other thermodynamic quantities). The
dynamics of this procedure is distinct from fixed weight updates. The
probability for a configuration to be sampled depends on a number of unusual
quantities, which are explained in this paper. This has been overlooked in
recent literature, where the method is advertised for the calculation of
canonical expectation values. We illustrate these points for the $2d$ Ising
model. In addition, we proof a previously conjectured equation which relates
microcanonical expectation values to the spectral density.
