论文标题
$ \nablaφ$接口的吉布斯采样器的光谱差距和截止现象具有凸电势
Spectral gap and cutoff phenomenon for the Gibbs sampler of $\nablaφ$ interfaces with convex potential
论文作者
论文摘要
我们考虑Gibbs采样器或与$ \ Mathbb {r}^n $相关的热浴动力学，描述了具有凸电势的$ \nablaφ$接口。在对潜力的最小化假设下，我们发现该过程的光谱差距始终由$ \ mathrm {gap} _n = 1- \ cos（π/n）$给出，并且对于所有$ε\ in（0,1）$，它的$ε$ mmixing时间满足$ t_n（sim）\ sim \ sim \ sim \ sim \ sim \ sim \ nim n} {2 \ mathrm {gap} _n} $ as $ n \ to \ infty $，从而建立了截止现象。结果揭示了普遍的行为，因为它们不取决于潜力的选择。
We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on $\mathbb{R}^N$ describing $\nablaφ$ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by $\mathrm{gap}_N=1-\cos(π/N)$, and that for all $ε\in(0,1)$, its $ε$-mixing time satisfies $T_N(ε)\sim \frac{\log N}{2\mathrm{gap}_N}$ as $N\to\infty$, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential.
