Quantitative approximate independence for continuous mean field Gibbs measures
Quantitative approximate independence for continuous mean field Gibbs measures
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DOI：
10.1214/22-ejp743
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发表时间：
2021-05
影响因子：
1.4
通讯作者：
D. Lacker
中科院分区：
文献类型：
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作者：
D. Lacker
许多具有平均场相互作用的吉布斯测量已知是混沌的，这意味着$n$ -粒子系统中任何$k$粒子的集合都是渐近独立的，就像$n\to\infty$与$k$固定或$k=o(n)$一样。本文对欧几里得空间上具有对偶相互作用的连续吉布斯测度的这一概念进行了量化，主要的例子是由凸相互作用和均匀凸约束势控制的系统。$k$粒子的边际定律与其极限积测度之间的距离为$O((k/n)^{c \wedge 2})$, $c$与温度的平方成正比。在高温情况下，这改进了基于熵次可加性的先前结果，最多产生$O(k/n)$。正如一个高斯的例子所证明的那样，bound $O((k/n)^2)$不能被改进。结果是非渐近的，距离通过相对Fisher信息、相对熵或平方二次Wasserstein度量来量化。该方法依赖于极限测度的先验函数不等式，用于根据$(k+1)$ -粒子距离推导$k$ -粒子距离的估计。
Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of $k$ particles in the $n$-particle system are asymptotically independent, as $n\to\infty$ with $k$ fixed or perhaps $k=o(n)$. This paper quantifies this notion for a class of continuous Gibbs measures on Euclidean space with pairwise interactions, with main examples being systems governed by convex interactions and uniformly convex confinement potentials. The distance between the marginal law of $k$ particles and its limiting product measure is shown to be $O((k/n)^{c \wedge 2})$, with $c$ proportional to the squared temperature. In the high temperature case, this improves upon prior results based on subadditivity of entropy, which yield $O(k/n)$ at best. The bound $O((k/n)^2)$ cannot be improved, as a Gaussian example demonstrates. The results are non-asymptotic, and distance is quantified via relative Fisher information, relative entropy, or the squared quadratic Wasserstein metric. The method relies on an a priori functional inequality for the limiting measure, used to derive an estimate for the $k$-particle distance in terms of the $(k+1)$-particle distance.