Convergence of metric graphs and energy forms

Abstract

In this paper, we begin with clarifying spaces obtained as limits of sequences of finite networks from an analytic point of view, and we discuss convergence of finite networks with respect to the topology of both the Gromov-Hausdorff distance and variational convergence called Γ-convergence. Relevantly to convergence of finite networks to infinite ones, we investigate the space of harmonic functions of finite Dirichlet sums on infinite networks and their Kuramochi compactifications.

Criticality theory for Schrödinger operators on graphs

“…The following characterization is shown in the continuum setting [25]. For Laplace type operators on graphs (with non-negative potentials) a similar results are found in [13,14,18].…”

Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

“…The Kuramochi p-compactification of an infinite weighted tree topologically coincides with the end compactification. We have another example, though it is simple, of the Kuramochi p-compactification of an infinite network, which is different from the end compactification (see [25], [19] for details).…”

Boundary representation of Dirichlet forms on discrete spaces

“…The evident reason for this is certainly that there is no obvious canonical geometric boundary of an infinite graph. Now, various works in recent years suggest that the Royden boundary of a transient graph can be considered as an analogue to the geometric boundary of a bounded set in Euclidean space [Kas10,GHK + 15,KLSW17]. Indeed, to develop the theory of transient graphs according to this point of view can be seen as the main driving force behind the considerations in [GHK + 15,KLSW17].…”