分支过程普适性的场论方法
Field-theoretic approach to the universality of branching processes
摘要：
分支过程被广泛用于从网络到神经元雪崩的现象建模。在一个大类的连续时间分支过程中，我们研究了瞬时人口规模，生存概率，预期雪崩持续时间，所谓的雪崩形状，$n$点关联函数和总雪崩大小的概率密度函数的时刻的时间标度。以往的研究表明，在某些观测的分支过程中使用概率参数的普遍性，但是，缺乏一个全面的描述。我们推导出描述这一过程的场论，并演示了如何使用它来计算相关的观测量及其在时间上的领先阶，揭示了人口规模的时刻的普遍性。我们的研究结果解释了为什么后代分布的第一和第二时刻是足够的，以充分的过程中的临界附近，无论潜在的后代分布。这一发现意味着分支过程是普遍的。我们用计算机模拟来说明我们的分析结果。
英文摘要：
Branching processes are widely used to model phenomena from networks to neuronal avalanching. In a large class of continuous-time branching processes, we study the temporal scaling of the moments of the instant population size, the survival probability, expected avalanche duration, the so-called avalanche shape, the $n$-point correlation function and the probability density function of the total avalanche size. Previous studies have shown universality in certain observables of branching processes using probabilistic arguments, however, a comprehensive description is lacking. We derive the field theory that describes the process and demonstrate how to use it to calculate the relevant observables and their scaling to leading order in time, revealing the universality of the moments of the population size. Our results explain why the first and second moment of the offspring distribution are sufficient to fully characterise the process in the vicinity of criticality, regardless of the underlying offspring distribution. This finding implies that branching processes are universal. We illustrate our analytical results with computer simulations.
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