Configuration space for random walk dynamics

Applied to statistical physics models, the random cost algorithm enforces a Random Walk (RW) in energy (or possibly other thermodynamic quantities). T he dynamics of this procedure is distinct from fixed weight updates. The pr obability 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 ca nonical expectation values. We illustrate these points for the 2d Ising mod el. In addition, we prove a previously conjectured equation which relates m icrocanonical expectation values to the spectral density.