Optimal Unimodular Matching
Abstract
We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.
Keywords
Cite
@article{arxiv.2407.03141,
title = {Optimal Unimodular Matching},
author = {Nathanaël Enriquez and Mike Liu and Laurent Ménard and Vianney Perchet},
journal= {arXiv preprint arXiv:2407.03141},
year = {2025}
}
Comments
58 pages, 19 figures. Improved overall presentation of the paper and added references