We revisit the topic of self-organized criticality (SOC) in simple statistical graph models, with the purpose of capturing essential processes leading to the emergence of macroscopic spacetime from the microscopic dynamics in loop quantum gravity. We performed a large set of simulations based on extensions of the frozen trivalent spin network (TSN) model explored previously by Ansari and Smolin. Their model mimicked the sandpile dynamics by the application of random vertex propagation rules in the TSN, leading to an SOC behavior in the distribution of the avalanche sizes, as well as a slowly expanding, two-dimensional dual (triangulated) space. Here we show that a growth scheme for the stochastic, slow external driving force, differing from the classical sandpile model, also resulted in power-law distributed avalanche sizes. Our simulations also produced expanding dual spaces, with two basic classes of evolution: one with power-law correlations in ‘space’ and ‘time’, and the other with ‘loitering’ and exponential phases. Our work expands the range of models in which critical states in the TSN may lead to expansion effects in the dual space, without fine-tuning.

我们在简单的统计图模型中重新审视自组织临界性 (SOC) 的主题，目的是从圈量子引力的微观动力学中捕捉导致宏观时空出现的基本过程。我们基于 Ansari 和 Smolin 先前探索的冻结三价自旋网络 (TSN) 模型的扩展进行了大量模拟。他们的模型通过在 TSN 中应用随机顶点传播规则来模拟沙堆动力学，导致雪崩大小分布中的 SOC 行为以及缓慢扩展的二维双（三角）空间。在这里，我们展示了随机的、缓慢的外部驱动力的增长方案，不同于经典的沙堆模型，也导致了幂律分布的雪崩大小。我们的模拟还产生了扩展的双空间，具有两种基本的进化类别：一种在“空间”和“时间”中具有幂律相关性，另一种具有“徘徊”和指数阶段。我们的工作扩展了模型的范围，其中 TSN 中的临界状态可能会导致双空间中的扩展效应，而无需微调。