Sharp mixing bounds for broader fixed-margin families

Derive sharp mixing-time estimates for the Snake algorithm for fixed-margin binary matrices beyond the permutation-matrix case, and characterize the margin-dependent behavior separating these estimates from the universal polynomial comparison bound.

Background

For arbitrary feasible row and column margins, the paper proves that the lazy Snake chain is rapidly mixing by comparison with the lazy swap chain, obtaining a universal polynomial spectral-gap and mixing-time bound. In the special permutation-matrix case, where both margin vectors equal the all-ones vector, the raw Snake chain is analyzed sharply and has total-variation mixing time of order Θ(√n log n).

The authors explicitly state that the universal estimate is not expected to capture the true behavior of the Snake chain for many margin families. The open problem is therefore to obtain sharp, margin-sensitive mixing-time results for classes of fixed-margin matrices broader than permutation matrices and to close the gap between the general comparison estimate and such margin-specific behavior.