A global spectral gap for Metropolis-adjusted Langevin algorithm with a uniformly randomized step size

Let $π(\mathrm{d} x)\propto e^{-U(x)}\,\mathrm{d} x$ on $\mathbb R^d$, where $0<m\leq L<\infty$, $mI_d\preceq\nabla^2U(x)\preceq LI_d$, and $κ=L/m$. It is known that, under warm-start assumptions, fixed-step Metropolis-adjusted Langevin algorithm (MALA) with properly tuned step size has mixing time