Action convergence of operators and graphs

Source abstract

Abstract 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.