  We analyze the hit-and-run algorithm for sampling uniformly from an isotropic
convex body $K$ in $n$ dimensions. We show that the algorithm mixes in time
$\tilde{O}(n^2/ \psi_n^2)$, where $\psi_n$ is the smallest isoperimetric
constant for any isotropic logconcave distribution, also known as the
Kannan-Lovasz-Simonovits (KLS) constant. Our bound improves upon previous
bounds of the form $\tilde{O}(n^2 R^2/r^2)$, which depend on the ratio $R/r$ of
the radii of the circumscribed and inscribed balls of $K$, gaining a factor of
$n$ in the case of isotropic convex bodies. Consequently, our result gives a
mixing time estimate for the hit-and-run which matches the state-of-the-art
bounds for the ball walk. Our main proof technique is based on an annealing of
localization schemes introduced in Chen and Eldan (2022), which allows us to
reduce the problem to the analysis of the mixing time on truncated Gaussian
distributions.
