We show that the logarithmic Sobolev inequality of the Boolean cube is stable under random monotone censoring. More precisely, if $A_n\subseteq \{0,1\}^n$ is chosen uniformly among all monotone subsets, then the logarithmic Sobolev constant of the censored walk on $A_n$ is of order $n$ with high probability. As a consequence, several analytic and probabilistic properties of the Boolean cube persist for a typical monotone subset: the censored semigroup is hypercontractive, the uniform measure on $A_n$ satisfies Gaussian concentration for Lipschitz observables, and the associated walk mixes in time $O(n\log n)$. The latter proves a conjectured mixing bound of Ding and Mossel for almost all monotone sets. The result is genuinely typical rather than universal. We construct monotone sets of density bounded away from zero whose logarithmic Sobolev constant is of order $n^2$. To prove the result, we establish a sharp logarithmic Sobolev inequality for Hamming caps and combine it with a harmonic extension argument transferring this inequality to monotone sets lying between nearby caps, together with a structural theorem of Korshunov on random monotone sets.

翻译：我们证明布尔立方体的对数索伯列夫不等式在随机单调删截下是稳定的。精确而言，若 $A_n\subseteq \{0,1\}^n$ 均匀取自所有单调子集，则 $A_n$ 上的删截游走的对数索伯列夫常数依大概率阶为 $n$。作为推论，布尔立方体的若干解析与概率性质对典型单调子集依然成立：删截半群是超压缩的，$A_n$ 上的均匀测度对Lipschitz可观测量满足高斯集中性，以及关联游走在时间 $O(n\log n)$ 内混合。后者证明了对于几乎所有单调集，Ding与Mossel的一个混合界猜想成立。该结果本质上是典型的而非普适的。我们构造了密度有界远离零的单调集，其对数索伯列夫常数阶为 $n^2$。为证明该结果，我们建立了汉明帽的尖刻对数索伯列夫不等式，并结合调和延拓论证将该不等式转移到位于相邻帽之间的单调集上，同时利用Korshunov关于随机单调集的结构定理。