The Structure of Typical Clusters in Large Sparse Random Configurations
The Structure of Typical Clusters in Large Sparse Random Configurations
复制标题
大稀疏随机配置中典型簇的结构
DOI：
10.1007/s10955-009-9728-y
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发表时间：
2008
影响因子：
1.6
通讯作者：
V. Sidoravicius
中科院分区：
文献类型：
--
作者：
J. Bertoin;V. Sidoravicius
这项工作的最初目的是对聚合数量有限的 Smoluchowski 凝固方程版本的最新结果提供概率解释。后者模拟介质中粒子浓度的确定性演化，其中粒子随着时间的推移成对合并，并且每个粒子只能执行给定数量的聚合。在适当的假设下，粒子的浓度随着时间趋于无穷大而收敛到某种程度，这与从两个祖先开始的高尔顿-沃森过程的总人口分布惊人地相似。粗略地说，配置模型是一种随机构造，旨在在具有预先描述的度数的一组顶点上生成典型的图。具体来说，为每个顶点附加一定数量的存根，然后均匀地随机地两两连接存根以在顶点之间创建边。在这项工作中，我们使用配置模型作为具有有限聚合的 Smoluchowski 凝固方程的随机对应物。我们建立了流体动力学类型极限定理，用于当顶点数量趋于无穷大时配置模型中团簇形状的经验测量。该极限是根据从两个祖先开始的高尔顿-沃森过程的分布给出的。
The initial purpose of this work is to provide a probabilistic explanation of recent results on a version of Smoluchowski’s coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors.Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices.In this work, we use the configuration model as the stochastic counterpart of Smoluchowski’s coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to ∞. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.