Before revision:  We show that from a warm start, the random walk mixes in O(n^7) steps where the log factors depend only on constants determined by the warm start and error parameters (and not on the dimension or number of constraints defining the body). This sampling algorithm thus offers improvement over the affine-invariant Dikin Walk for polytopes (which mixes in (mn) steps from a warm start) for applications in which m \gg n.
After revision:  We show that from a warm start, the random walk mixes in O(n^7) steps where the log factors depend only on constants determined by the warm start and (mn) steps from a warm start) for applications in which m \gg n.