Group-averaged Markov chains II: tuning of group action in finite state space
We study group-averaged Markov chains obtained by augmenting a $π$-stationary kernel $P$ with orbit kernels induced by a group action. We analyse the Gibbs ($G$), Metropolis--Hastings ($M$), and Barker ($B$) kernels, their sandwiches $QPQ$, and mixtures $\tfrac{1}{2}(P+Q)$, where $Q\in\{G,M,B\}$. Under suitable conditions, $M^t$ and $B^t$ converge blockwise to $G$. The projection chains of $GPG$ and $P$ coincide, while every sandwich $QPQ$ has absolute spectral gap no smaller than that of reversible $P$. For $GPG$, we derive an additive asymptotic-variance bound, prove monotonicity for $G$-invariant observables, and identify it as the Kullback--Leibler (KL) information projection of $P$ onto the $G$-invariant kernels. For a fixed orbit partition, the spectral and KL properties of $GPG$ reduce to those of a lower-dimensional orbit-space chain. Among Gibbs projections with a prescribed number of orbits, we identify the partition minimizing KL divergence to stationarity and characterize exact stationarity. Finally, alternating group projections converge at a rate determined by singular values of an overlap matrix and, in structured cases, can yield exact sampling with logarithmically many group actions. These results motivate tuning heuristics and yield polynomial mixing for a Curie--Weiss example in a regime where Glauber dynamics is exponentially slow.