论文标题
麦克斯韦（Maxwell）原理
A Maxwell principle for generalized Orlicz balls
论文作者
论文摘要
在[十二个de {f} inetti风格的结果导致搜索理论，Ann。研究H.PoincaréProbab。统计学家。 23（2）（1987），397--423]，Diaconis和Freedman研究了来自欧几里得单位球体的随机向量的低维投影，在高维度中研究了这些随机矢量的单个坐标看起来分别像高斯和指数随机变量。在随后的作品中，Rachev和Rüschendorf和Naor和Romik通过在$ \ ell_p^n $ Balls与$ P $归纳的高斯分布之间建立联系，从而统一了这些结果。在本文中，我们在显着概括和统一的环境中研究了类似的问题，查看随机矢量的低维投影，均匀分布在形式的集合上\ [b_ {ϕ，t}^n：= \ big \ \ \ \ {（s_1，s_1，\ ldots，\ ldots，s_n） = 1}^n ϕ（s_i）\ leq t n \ big \}，\]其中$ ϕ：\ mathbb {r} \ to [0，\ infty] $是潜在的（包括orlinicz函数的情况）。我们的方法与Rachev-rüschendorf和Naor-Romik不同，基于Cramér定理的定量版本和Gibbs条件原则的较大偏差的观点，提供了一个自然框架，在$ p $ p $ p $ generalized的高斯分布中，同时散布了$ planse play n plans $ plans $ n plans $ n n $ n n $ n of。我们发现有一个关键参数$ t _ {\ mathrm {crit}} $，其中预测行为存在相变：for $ t> t _ {\ mathrm {crit}} $从$ b_ {ϕ，t}，t}^n Simpl ul bel yequave的随机点所取得的随机点所示的坐标。 t _ {\ mathrm {crit}} $ gibbs调节原理起作用，这里有一个参数$β_T> 0 $（反向温度），因此坐标是根据与$ e^{-β_T（s）} $的密度成正比的密度分布的。
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincaré Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Rüschendorf and Naor and Romik unified these results by establishing a connection between $\ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form \[B_{ϕ,t}^N := \Big\{(s_1,\ldots,s_N)\in\mathbb{R}^N : \sum_{ i =1}^Nϕ(s_i)\leq t N\Big\},\] where $ϕ:\mathbb{R}\to [0,\infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-Rüschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cramér's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $\ell_p^N$ balls. We find that there is a critical parameter $t_{\mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{\mathrm{crit}}$ the coordinates of random points sampled from $B_{ϕ,t}^N$ behave like uniform random variables, but for $t \leq t_{\mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $β_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -β_tϕ(s)}$.
