Quantitative speed-up bound for the reflected Burnside process

Prove a quantitative bound on the speed-up achieved by the reflected Burnside process, whose transition kernel is obtained by composing the lumped Burnside process on integer partitions with the deterministic partition-transpose map.

Background

The paper introduces the reflected Burnside process on the set of integer partitions of n by first transposing the current partition and then applying one step of the lumped Burnside process. Its transition kernel is Q = ΠP, where P is the lumped Burnside kernel and Π is the permutation matrix corresponding to partition transposition. The process has the uniform distribution on partitions as its stationary distribution and is reported empirically to mix substantially faster than the original lumped Burnside process.

Although simulations suggest that the mixing time of the reflected process grows logarithmically with n and remains small for very large n, the paper does not establish a theorem quantifying this improvement. The unresolved problem is therefore to derive a rigorous bound measuring the speed-up produced by the deterministic transpose jump.