Circular Planar Electrical Networks II: Positivity Phenomena

Abstract: Curtis-Ingerman-Morrow characterize response matrices for circular planar electrical networks as symmetric square matrices with row sums zero and non-negative circular minors. In this paper, we study this positivity phenomenon more closely, from both algebraic and combinatorial perspectives. Extending work of Postnikov, we introduce electrical positroids, which are the sets of circular minors which can simultaneously be positive in a response matrix. We give a self-contained axiomatic description of these electrical positroids. In the second part of the paper, we discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy. We investigate the clusters in this algebra, building off of initial work by Kenyon-Wilson, using an analogue of weak separation, as was originally introduced by Leclerc-Zelevinsky.