We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös–Rényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.

中文翻译：

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在Erdös-Rényi随机图中逼近三角形的累积生成函数

我们研究了“边缘三角模型”的压力，该压力等于Erdös-Rényi随机图中三角形的累积生成函数。研究涉及在体积增加的有限图中的种群动力学方法，以及在无限体积限制下产生的石墨烯变分问题的离散化。结果，我们在参数空间中找到一条曲线，其中发生了一步复制对称中断过渡。在具有对称性的情况下，一个非常接近等分图的结构的石墨烯很好地描述了在对称对称破坏阶段对大型图进行采样的过程。