论文标题
吉布斯分区：综合相图
Gibbs partitions: a comprehensive phase diagram
论文作者
论文摘要
我们研究吉布斯分区模型，也称为组成方案。我们的主要结果全面地描述了它们的相图，包括从Stufler（2018，随机结构\＆算法）中描述的相位过渡到一个新的致密状态，其特征是线性数量的组件，其较小订单的波动量，由$α$稳定的法律量化，以$ 1 <α\α\ le 2 $ 2 $ 2 $ 2 $ 2 $。我们证明了一个流程对应于组件大小的过程的功能缩放限制，并讨论了对极端组件大小的应用程序。在过渡时，我们观察到两个渐近形状的混合物。我们还处理扩展的组成方案，并在稀释状态下证明了局部限制定理，限制法律与$α$稳定的法律有关，价格为$ 0 <α<1 $。我们通过点过程限制描述了最大组件的渐近大小。
We study Gibbs partition models, also known as composition schemes. Our main results comprehensively describe their phase diagram, including a phase transition from the convergent case described in Stufler (2018, Random Structures \& Algorithms) to a new dense regime characterized by a linear number of components with fluctuations of smaller order quantified by an $α$-stable law for $1< α\le 2$. We prove a functional scaling limit for a process whose jumps correspond to the component sizes and discuss applications to extremal component sizes. At the transition we observe a mixture of the two asymptotic shapes. We also treat extended composition schemes and prove a local limit theorem in a dilute regime with the limiting law being related to an $α$-stable law for $0< α< 1$. We describe the asymptotic size of the largest components via a point process limit.
