Circular planar electrical networks: Posets and positivity

Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We first construct a poset EPn of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. We then answer various enumerative questions related to EPn, adapting methods of Callan and Stein-Everett. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of Postnikov, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy.