On two proofs of $d^2$ 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, $\bar{\nu }$-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 $\tilde{O}(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\tilde{O}(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger ${\chi }^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee–Sidford metric, this yields an $\tilde{O}(d^2)$ mixing bound in ${\chi }^2$-divergence, improving on the previous $\tilde{O}(d^{9\mathrm{/}4})$ bound.