Uniform positivity of the tau invariant
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.
Related
- Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven
- From the Optimal L2-Estimates (p,n) and (n,p)-Forms to Uniform Griffiths p-Positivity via Convolution and Restriction
- Individual and Uniform Eventual Positivity of Self-Adjoint Semigroups
- Uniform RC-positivity of tangent bundles
- A zero-one law for swap-invariant events in uniform spanning forests
BibTeX
@misc{wang2026uniform,
title = {{Uniform positivity of the tau invariant}},
author = {Ruihua Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04994},
eprint = {2610.04994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04994},
}
RIS
TY - UNPB
TI - Uniform positivity of the tau invariant
AU - Ruihua Wang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.04994
UR - https://arxiv.org/abs/2610.04994
AB - 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.
ER -