Strongly coupled Dyson–Schwinger equations generate infinite power series of running coupling constants together with Feynman diagrams with increasing loop orders as coefficients. Theory of graphons for sparse graphs can address a new useful approach for the study of graph limits of sequences of partial sums corresponding to these infinite power series in the context of Feynman graphons and the cut-distance topology. Graphon models enable us to associate some new analytic graphs to non-perturbative solutions of Dyson–Schwinger equations. Homomorphism densities of Feynman graphons provide a new way of analyzing non-perturbative phase transitions. We explain the structures of topological renormalization quotient Hopf algebras of Feynman graphons which encode gauge symmetries Hopf ideals in the context of the weakly isomorphic equivalence classes corresponding to the Slavnov–Taylor / Ward–Takahashi elements. We apply Feynman graphon representations of combinatorial Dyson–Schwinger equations underlying the Connes–Kreimer renormalization Hopf algebra to construct a new class of Banach bundles for the study of the dynamics of non-perturbative phases in strongly coupled gauge field theories.

强耦合戴森-施温格方程生成运行耦合常数的无穷幂级数以及费曼图，循环阶数作为系数增加。稀疏图的图子理论可以解决一种新的有用方法，用于研究在费曼图子和切割距离拓扑的背景下与这些无穷幂级数相对应的部分和序列的图极限。Graphon 模型使我们能够将一些新的解析图与 Dyson-Schwinger 方程的非微扰解相关联。费曼图子的同态密度提供了一种分析非微扰相变的新方法。我们在对应于 Slavnov-Taylor / Ward-Takahashi 元素的弱同构等价类的背景下解释了费曼图子的拓扑重整化商 Hopf 代数的结构，该代数编码规范对称性 Hopf 理想。我们应用基于 Connes-Kreimer 重整化 Hopf 代数的组合 Dyson-Schwinger 方程的费曼图子表示来构建一类新的 Banach 丛，用于研究强耦合规范场理论中非微扰相的动力学。