Action convergence of operators and graphs. (17th February 2022)

• Record Type: Journal Article
• Authors: Backhausz, Ágnes; Szegedy, Balázs
• Is Part Of: Canadian journal of mathematics. Volume 74:Number 1(2022)
• Page Start: 72
• Page End: 121
• Publication Date: 2022-02-17
• Subjects: 05C50; Graph limits -- operator -- random matrix; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X2000070X
• Languages: English
• ISSNs: 0008-414X

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.