Uniform positivity of the tau invariant

Source abstract

We prove the Baker--Rumely conjecture that the tau invariant of a metrized graph admits a positive lower bound proportional to its total length, with an absolute constant. We also construct simple cubic metrized graphs whose normalized tau invariants tend to $59/7260<1/108$, disproving the proposed universal constant $1/108$. The lower bound is independent of the genus, the number of edges, and the distribution of edge lengths. Its proof combines a second-moment inequality for Euclidean lattices with a partition of the edge coordinates of a cycle lattice into three independent sets. The counterexamples have only two edge lengths and admit an elementary resistance calculation. Through the tropical moment identity, the lower bound also gives a uniform estimate for the non-archimedean terms in height formulas for Jacobians.