Configuration Space for Random Walk Dynamics

TL;DR

This paper investigates the dynamics of random walk algorithms in statistical physics, revealing overlooked aspects of their sampling probabilities and proving a key relation between microcanonical expectations and spectral density.

Contribution

It clarifies the sampling dynamics of the random cost algorithm and proves a conjectured relation in statistical physics models.

Findings

Sampling probability depends on unusual quantities

The dynamics differ from fixed weight updates

Proved a relation between microcanonical expectations and spectral density

Abstract

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.