2018/11/01 by Agnes Backhausz, Backhausz, Agnes, Balazs Szegedy +1 · 1 voice · 8 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #math.CO #math.FA #math.PR
paper · pdf · doi:10.48550/arxiv.1811.00626
arxiv created 2018/11/01 · arxiv published 2018/11/01 · arxiv updated 2018/11/05
We present a new approach to graph limit theory which unifies and generalizes the two most well developed directions, namely dense graph limits (even the more general Lp 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)→ L1(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, Lp graphons and graphings (known from graph limit theory) are special examples for graphops. We describe a new point of view on random matrix theory using our operator limit framework.