We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.

翻译：本文首次推导了随机游走Metropolis算法在$\mathbb{R}^d$上任意提议方差下的谱间隙显式界，当适当缩放时，该界恢复了对适当正则不变分布的正确维数依赖性$d^{-1}$。我们还获得了一类广泛模型的${\rm L}^2$-混合时间的显式界。在推导这些结果的过程中，我们改进了等周轮廓不等式的使用以获得电导轮廓界，这还使得能够在更广泛的模型类中导出显式界。此外，我们针对预条件Crank–Nicolson马尔可夫链获得了类似结果，在适当假设下得到了与维数无关的界。