向量乘法凝聚中的凝胶化与多类型泊松分支过程中的灭绝
Gelation in Vector Multiplicative Coalescence and Extinction in Multi-Type Poisson Branching Processes
AI总结：
本文建立了向量乘法凝聚过程与多类型泊松分支过程之间的等价关系，据此给出凝胶化的快速证明及灭绝概率的级数表达式，并利用随机图推导修正Smoluchowski方程的解。
摘要：
在本文中，我们提出了多类型（向量）乘法凝聚过程与具有泊松后代分布的多类型分支过程之间的一种新联系。更具体地说，我们证明了控制向量乘法凝聚过程中凝胶化现象的方程，与给出相应多类型泊松分支过程灭绝概率的方程是等价的。然后，我们利用这一联系进行两个应用，每个方向各一个。第一个是向量乘法凝聚过程中凝胶化的一个新的快速证明，第二个是多类型泊松分支过程灭绝概率的一个新的级数表达式。我们还利用随机图对修正的Smoluchowski凝聚方程的解给出了新的推导，该方程描述了向量乘法凝聚过程。
英文摘要：
In this note, we present a novel connection between a multi-type (vector) multiplicative coalescent process and a multi-type branching process with Poisson offspring distributions. More specifically, we show that the equations that govern the phenomenon of gelation in the vector multiplicative coalescent process are equivalent to the equations that yield the extinction probabilities of the corresponding multi-type Poisson branching process. We then leverage this connection with two applications, one in each direction. The first is a new quick proof of gelation in the vector multiplicative coalescent process, and the second is a new series expression for the extinction probabilities of the multi-type Poisson branching process. We also use random graphs to give a new derivation of the solution to the modified Smoluchowski coagulation equations, which describe the vector multiplicative coalescent process.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
