A Tight Bound on Localization of Electrical Flows
Data Structures and Algorithms
2026-05-26 v1 Discrete Mathematics
Probability
Abstract
We prove that for any unweighted graph on n vertices the L1 norm of a unit electric current between the endpoints of a random edge is at most 2 log n. Furthermore, we show that on any weighted graph the spectral norm of the entry-wise absolute value of the symmetric transfer-current matrix is at most 2 log n. This bound is tight up to constants and improves the O(log^2 n) bound from [Schild-Rao-Srivastava, SODA '18]. The initial proofs were generated by OpenAI's ChatGPT 5.5 Pro; the authors have verified and rewritten them to enhance readability and provide additional context.
Cite
@article{arxiv.2605.24130,
title = {A Tight Bound on Localization of Electrical Flows},
author = {Ori Gurel-Gurevich and Asaf Nachmias and Sushant Sachdeva},
journal= {arXiv preprint arXiv:2605.24130},
year = {2026}
}