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We introduce higher rank inner products on real and complex vector spaces and study their corresponding Voronoi tilings. We use the framework to describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings of discrete sets., Comment: 65 pages, 12 figures. Comments welcome!

In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator., Comment: 29 pages, 1 figure

We provide an introductory review of some topics in spectral theory of Laplacians on metric graphs. We focus on three different aspects: the trace formula, the self-adjointness problem and connections between Laplacians on metric graphs and discrete graph Laplacians., Comment: to appear in Internationale Mathematische Nachrichten; 15 pages, 2 figures

The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions. As a byproduct of our approach, we obtain other results of independent interest. In particular, we introduce higher rank canonical compactifications of fans and polyhedral spaces and use them to define the moduli space of higher rank tropical curves. Moreover, we develop the first steps of a function theory in higher rank non-Archimedean, hybrid, and tame analysis. Furthermore, we establish the convergence of the Laplace operator on metric graphs toward the tropical Laplace operator on limit tropical curves in the corresponding moduli spaces, leading to new perspectives in operator theory on metric graphs. Our result on the Arakelov Green function is inspired by the works of several authors, in particular those of Faltings, de Jong, Wentworth and Wolpert, and solves a long-standing open problem arising from the Arakelov geometry of Riemann surfaces. The hybrid layered behavior close to the boundary of moduli spaces is expected to be a broad phenomenon and will be explored in our forthcoming work., Comment: 230 pages, 26 figures, comments welcome