Abstract
We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the edge weights are positive. Under suitable conditions, these lead to comparison of series of matrix powers which express those kernels. We show that the notions of transience and recurrence extend by analytic continuation to the complex-weighted case even when the network is infinite. Thus, a variety of methods known for Markov chains extend to that setting.
Highlights
We have continued the research by the first author [12,13,22] on networks with complex edge-weights
The latter are admittances parametrized by a complex number s with positive real part
Building upon basic results for finite networks, our main focus is on infinite, locally finite ones
Summary
In the present paper we consider exclusively the case s P Hr , the right half plane consisting of all complex numbers with Re s ą 0 While this is a technical assumption which is crucial for the present approach, it is typical in network theory: the admittance (1) is a positive-real function, that is, Re ρs px, yq ą 0 when Re s ą 0 ; see [12,14–17]. It is convenient to work with the inverse of effective impedance, that is, the effective admittance, which corresponds to the total amount of current in the electrical network In this context, we provide first comparisons of associated power series with analogous ones for reversible Markov chains.
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