  We present a new random walk for uniformly sampling high-dimensional convex
bodies. It achieves state-of-the-art runtime complexity with stronger
guarantees on the output than previously known, namely in R\'enyi divergence
(which implies TV, $\mathcal{W}_2$, KL, $\chi^2$). The proof departs from known
approaches for polytime algorithms for the problem -- we utilize a stochastic
diffusion perspective to show contraction to the target distribution with the
rate of convergence determined by functional isoperimetric constants of the
stationary density.
