ar X iv : 0 80 6 . 38 81 v 1 [ m at h . O A ] 2 3 Ju n 20 08 OPERATOR THEORY OF ELECTRICAL RESISTANCE NETWORKS

An electrical resistance network (ERN) is a weighted graph (G, μ). The edges are weighted by a function μ and interpreted as resistors of possibly varying strengths. The effective resistance metric R(x, y) is the natural notion of distance between two vertices x, y in the ERN. We discuss how R can be formulated in terms of an energy form E defined for functions on the vertices, a dissipation form D defined for functions on the edges, the graph Laplacian ∆ defined for functions on the vertices, or by probabilistic methods. The Dirichlet energy form E is similar to those studied in [FŌT94], but certain technical differences yield a strikingly different theory. We construct an embedding of the ERN (equipped with resistance metric) into a Hilbert space under which each vertex x corresponds to a vector vx. Such a vector solves the equation ∆vx = δx − δo, where δx denotes a Dirac mass at x and o is some fixed reference vertex. The central object of study is the Hilbert space HE of functions of finite energy E. We show that {vx} is a reproducing kernel for this space, and exploit this to obtain a detailed description of the structure of HE . In particular, we show that ERNs which support nonconstant harmonic functions of finite energy have a certain type of 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 of bounded (resp. nonnegative) harmonic functions, but without any probabilistic assumptions. In the process, we construct a dense space of “smooth” functions of finite energy and obtain a Gel’fand triple for HE , thus providing a method for performing a mildly restricted form of Fourier analysis on a general ERN. We also analyze the Hilbert space of functions defined on the edges and are able to solve the problem of making such functions compatible (in the sense of potential theory) with functions in HE via the use of an operator which implements Ohm’s law, and its adjoint. In addition, these operators allow us to convert a Dirichlet problem on the vertex space to a Dirichlet problem on the edge space, and vice versa; this is exploited to solve boundary-value problems and establish some existence results. We provide a couple of candidates for frames (and dual frames) when working with an infinite ERN. A key to our analysis is the observation that the spectral representation for the graph Laplacian ∆ on HE is drastically different from the corresponding representation on l. Since the ambient Hilbert space HE is defined by the energy form, many interesting phenomena arise which are not present in l; we highlight many examples and explain why this occurs. In particular, we show how the boundary of an ERN corresponds to the deficiency indices of ∆ as an operator on HE , and hence how the boundary is detected by the operator theory of HE but not l . The Hilbert space framework allows us to study properties of the graph Laplacian and its associated transfer operator, including boundedness, compactness, essential self-adjointness and other properties. We examine flows on the network induced by currents; and the probabilistic interpretation these flows yield. This leads to the notion of forward-harmonic functions, for which we also provide a characterization in terms of a boundary representation. Using our results we establish precise bounds on correlations in the Heisenberg model for quantum spin observables, and we improve earlier results of R. T. Powers. Our focus is on the quantum spin model on the rank-3 lattice, i.e., the ERN with Z as vertices and with edges between nearest neighbors. This is known as the problem of long-range order in the physics literature, and refers to KMS states (Kubo-Martin-Schwinger) on the C-algebra of the model. OPERATOR THEORY OF ELECTRICAL RESISTANCE NETWORKS 3