We present a new approach to graph limit theory that unifies and generalizes the two most well-developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini–Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called P-operators) of the form $L^\infty (\Omega )\to L^1(\Omega )$ for probability spaces $\Omega $ . We introduce a metric to compare P-operators (for example, finite matrices) even if they act on different spaces. We prove a compactness result, which implies that, in appropriate norms, limits of uniformly bounded P-operators can again be represented by P-operators. We show that limits of operators, representing graphs, are self-adjoint, positivity-preserving P-operators called graphops. Graphons, $L^p$ graphons, and graphings (known from graph limit theory) are special examples of graphops. We describe a new point of view on random matrix theory using our operator limit framework.

我们提出了一种新的图极限理论方法，它统一并概括了两个最发达的方向，即密集图极限（甚至更一般的 $L^p$ 极限）和 Benjamini-Schramm 极限（即使在更强的局部全局环境）。我们通过例子说明这个新框架为中等密度图提供了一个丰富的极限理论和自然极限对象。此外，它为概率空间 $\Omega $ 的形式 为 $L^\infty (\Omega )\to L^1(\Omega )$ 的有界算子（称为P算子）提供了极限理论。我们引入一个度量来比较P - 运算符（例如，有限矩阵），即使它们作用于不同的空间。我们证明了一个紧致性结果，这意味着在适当的规范中，一致有界的P算子的极限可以再次由P算子表示。我们表明，表示图的算子的极限是自伴随的、保持正性的P算子，称为 graphops。Graphons、 $L^p$ graphons 和 graphings（从图极限理论中知道）是 graphops 的特殊例子。我们使用我们的算子限制框架描述了随机矩阵理论的新观点。