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Applied to statistical physics models, the random cost algorithm enforces aRandom Walk (RW) in energy (or possibly other thermodynamic quantities). Thedynamics of this procedure is distinct from fixed weight updates. Theprobability for a configuration to be sampled depends on a number of unusualquantities, which are explained in this paper. This has been overlooked inrecent literature, where the method is advertised for the calculation ofcanonical expectation values. We illustrate these points for the $2d$ Isingmodel. In addition, we proof a previously conjectured equation which relatesmicrocanonical expectation values to the spectral density.Comment: Various minor changes, appendix added, Fig. 2 droppe

Configuration space for random walk dynamics