The Dual Burnside Process

Abstract: The Burnside process is a classical Markov chain for sampling uniformly from group orbits. We introduce the dual Burnside process, obtained by interchanging the roles of group elements and states. This dual chain has stationary law $\pi(g)\propto |X_g|$, is reversible, and admits a matrix factorization $Q=AB$, $K=BA$ with the classical Burnside kernel $K$. As a consequence the two chains share all nonzero eigenvalues and have mixing times that differ by at most one step. We further establish universal Doeblin floors, orbit and conjugacy-class lumpings, and transfer principles between $Q$ and $K$. We analyze explicit examples: the value-permutation model $S_k$ acting on $[k]<sup>n$ and the coordinate-permutation model $S_n$ acting on $[k]<sup>n$. These results show that the dual chain provides both a conceptual mirror to the classical Burnside process and practical advantages for symmetry-aware Markov chain Monte Carlo.