Operator theory of electrical resistance networks

Abstract

A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance) metric $R$. We embed the resistance network into the Hilbert space ${\mathcal{H}}_{\mathcal{E}}$ of functions of finite energy. We use the resistance metric to study ${\mathcal{H}}_{\mathcal{E}}$, and vice versa and show that the embedded images of the vertices $\{v_x\}$ form a reproducing kernel for this Hilbert space. We also obtain a discrete version of the Gauss-Green formula for resistance networks and show that resistance networks which support nonconstant harmonic functions of finite energy have a certain type of \emph{boundary}. We obtain an analytic boundary representation for the harmonic functions of finite energy in a sense analogous to the Poisson or Martin boundary representations, but with different hypotheses, and for a different class of functions. In the process, we construct a dense space of "smooth" functions of finite energy and obtain a Gel'fand triple for ${\mathcal{H}}_{\mathcal{E}}$. This allows us to represent the resistance network as a system of Gaussian random variables indexed by vertices. We also study the spectral representation for $\Delta$ on ${\mathcal{H}}_{\mathcal{E}}$ and show how nonzero defect entails a nontrivial boundary. All of the above are are detected by the operator theory of ${\mathcal{H}}_{\mathcal{E}}$ but not ${\ell }^2$. Our results apply to the Heisenberg model for the isotropic ferromagnet, improving earlier results of R. T. Powers on the problem of long-range order (in reference to KMS states on the $C^{\ast }$-algebra of the model).

330 pages, numerous figures

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Cited by in corpus (13)

• Persistent Laplacians: properties, algorithms and implications
• Non-Commutative Resistance Networks
• A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks
• Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics
• Spectral Theory for Discrete Lapacians
• A Hilbert space approach to effective resistance metric
• Self-adjoint extensions of network Laplacians and applications to resistance metrics
• Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian
• Analysis of unbounded operators and random motion
• Spectral duality for a class of unbounded operators
• Toeplitz Operators in Hilbert Space over Graphs
• A duality theory for unbounded Hermitian operators in Hilbert space
• A discrete Hopf-Rinow-theorem