Windowed thinning and query complexity for the bouncy particle and Zigzag samplers

Let μ(d x)\propto e^{-U(x)} d x on \R^d, where U is m-strongly convex and L-smooth, and denote by κ=L/m the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error \varepsilon, the expected query counts are O(κ^{1/2}d,(d\logκ+\log\frac1\varepsilon)) gradient queries for the bouncy particle sampler and O(κd^{1/4}(d\logκ+\log\frac1\varepsilon)) full-gradient equivalents for Zigzag, where d coordinate-partial queries count as one equivalent.