  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.
