Let d ≥ 3 be fixed and G be a large random d-regular graph on n vertices. We show that if n is large enough then the entry distribution of every almost eigenvector v of G (with entry sum 0 and normalized to have length √ n) is close to some Gaussian distribution N (0, σ) in the weak topology where 0 ≤ σ ≤ 1. Our theorem holds even in the stronger sense when many entries are looked at simultaneously in small random neighborhoods of the graph. Furthermore, we also get the Gaussianity of the joint distribution of several almost eigenvectors if the corresponding eigenvalues are close. Our proof uses graph limits and information theory. Our results have consequences for factor of i.i.d. processes on the infinite regular tree.
Abstract. In a random graph model introduced in [1] we give the joint asymptotic distribution of weights and degrees and prove scale-free property for the model. Moreover, we determine the asymptotics of the maximal weight and the maximal degree.
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 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-operators) of the form ∞ (Ω) → 1 (Ω) for probability spaces Ω. We introduce a metric to compare-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-operators can again be represented byoperators. We show that limits of operators representing graphs are self-adjoint, positivity-preserving-operators called graphops. Graphons, 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.
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