We study Glauber dynamics for sampling from discrete distributions $\mu$ on the hypercube $\{\pm 1\}^n$. Recently, techniques based on spectral independence have successfully yielded optimal $O(n)$ relaxation times for a host of different distributions $\mu$. We show that spectral independence is universal: a relaxation time of $O(n)$ implies spectral independence. We then study a notion of tractability for $\mu$, defined in terms of smoothness of the multilinear extension of its Hamiltonian -- $\log \mu$ -- over $[-1,+1]^n$. We show that Glauber dynamics has relaxation time $O(n)$ for such $\mu$, and using the universality of spectral independence, we conclude that these distributions are also fractionally log-concave and consequently satisfy modified log-Sobolev inequalities. We sharpen our estimates and obtain approximate tensorization of entropy and the optimal $\widetilde{O}(n)$ mixing time for random Hamiltonians, i.e. the classically studied mixed $p$-spin model at sufficiently high temperature. These results have significant downstream consequences for concentration of measure, statistical testing, and learning.

翻译：我们研究从超立方体 $\{\pm 1\}^n$ 上的离散分布 $\mu$ 中采样的格劳伯动力学。近年来，基于谱独立性的技术已成功为多种不同分布 $\mu$ 获得最优的 $O(n)$ 松弛时间。我们证明谱独立性具有普适性：$O(n)$ 的松弛时间意味着谱独立性。随后，我们研究 $\mu$ 的可处理性概念，该概念定义为其哈密顿量（$\log \mu$）在 $[-1,+1]^n$ 上的多重线性扩展的光滑性。我们证明对于此类 $\mu$，格劳伯动力学具有 $O(n)$ 的松弛时间，并利用谱独立性的普适性得出结论：这些分布也是分数对数凹的，从而满足修正的对数-索博列夫不等式。我们进一步改进估计，获得了熵的近似张量化以及随机哈密顿量（即经典研究的高温混合 $p$-自旋模型）的最优 $\widetilde{O}(n)$ 混合时间。这些结果对测度集中、统计检验和学习具有重要的下游影响。