On two proofs of d2 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, barnu-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 widetilde O(d2) mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an widetilde O(d4) mixing bound for sampling from truncated PSD cones. Our second result establishes stronger chi2-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an widetilde O(d2) mixing bound in chi2-divergence, improving on the previous widetilde O(d9/4) bound.