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 $π(g)\propto |X_g|$, 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, exact stabilizer/fixed-set quotient pairs, and transfer principles between $Q$ and $K$. We analyze the explicit examples of the value-permutation model $S_k$ acting on $[k]^n$ and the coordinate-permutation model $S_n$ acting on $[k]^n$. In the value-permutation model, for fixed $k\ge3$, the dual fixed-symbol-set quotient has $2^k-k-1$ states, independent of $n$, preserves the full nonzero spectrum, and has limiting nontrivial spectral radius $1/2$. These results show that the dual chain provides both a conceptual mirror to the classical Burnside process and a genuinely useful compression mechanism for symmetry-aware Markov chain Monte Carlo.