论文标题
Orlicz球的最大熵原理和体积特性
The maximum entropy principle and volumetric properties of Orlicz balls
论文作者
论文摘要
我们研究了Orlicz空间中球的精确渐近体积，并表明当环境空间的尺寸倾向于无限时，两个Orlicz球的交点的体积会经历相变。这概括了Schechtman和Schmuckenschläger的结果[Gafa，数学中的讲义。 1469（1991），174--178]，用于$ \ ell_p^d $ -Balls。作为另一个应用程序，我们确定$ 2 $ -Concave Orlicz Spaces $ \ ell_m^d $的精确渐近量比。我们的方法取决于统计力学和大偏差理论的思想，更确切地说，是非相互作用粒子的最大熵或吉布斯原理，并提出了这种自然方法和新的透视图，以表明这种几何和体积问题。特别是，我们的方法解释了与$ \ ell_p^d $ -balls相关的问题中的$ p $笼统的高斯分布如何发生，当Orlicz函数为$ m（t）= | t |^p $时，它们是Orlicz球。
We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschläger [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the $p$-generalized Gaussian distribution occurs in problems related to the geometry of $\ell_p^d$-balls, which are Orlicz balls when the Orlicz function is $M(t) = |t|^p$.
