Action convergence of operators and graphs. (17th February 2022) - Backhausz, Ágnes; Szegedy, Balázs
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^∞ (Ω )→ L^1(Ω ) for probability spaces Ω . 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.