Tropical moments of tropical Jacobians

Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, the tau invariant, and the total length of a metric graph. We argue that this linear relation is a non-archimedean analogue of a recent remarkable identity established by Wilms for invariants of compact Riemann surfaces. We also relate our work to the computation of heights attached to principally polarized abelian varieties.