On two proofs of d squared mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, ν-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an O(d 2) mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an O(d 4) mixing bound for sampling from truncated PSD cones. Our second result establishes stronger χ squared-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an O(d 2) mixing bound in χ squared-divergence, improving on the previous O(d 9/4) bound.