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    Large components in inhomogeneous random graphs
    (2025)
    Lienau, Matthias
    ;
    Schulte, Matthias
    This thesis studies two types of inhomogeneous random graphs, so-called rank-1 models and the weighted random connection model. Under suitable parameter choices, both random graphs exhibit a scale-free degree distribution as observed in real-world complex networks. For both models, we study the sizes of large components in the subcritical regime. In the rank-1 case, we also establish quantitative Poisson approximation results for cycle counts. The latter allow us to deduce the asymptotic distributions of the lengths of the shortest and of the longest cycle in the subcritical regime.
    Publicationtype: Doctoral Thesis
      74  91
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    Lower variance bounds and normal approximation of Poisson functionals in stochastic geometry
    (2024)
    Trapp, Vanessa
    ;
    Schulte, Matthias
    Lower bounds for variances are often required for deriving central limit theorems. In this thesis, a generalised reverse Poincaré inequality is established, which provides a lower variance bound for Poisson functionals and depends on the difference operator of some fixed order. To show how this lower variance bound can be used, three different applications from stochastic geometry are explored: statistics of spatial random graphs in Euclidean and hyperbolic space, Lp surface areas of random polytopes in the Euclidean unit ball and geometric functionals of excursion sets of Poisson shot noise processes.
    Publicationtype: Doctoral Thesis
      83  149