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 EPnEPn 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 EPnEPn, 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.