{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-number-of-optimal-matchings-for-euclidean","title":"The number of optimal matchings for Euclidean Assignment on the line","arxiv_id":"2101.04926","date":"2021-01-13","proceeding":null,"authors":["Sergio Caracciolo","Vittorio Erba","Andrea Sportiello"],"abstract":"We consider the Random Euclidean Assignment Problem in dimension $d=1$, with linear cost function. In this version of the problem, in general, there is a large degeneracy of the ground state, i.e. there are many different optimal matchings (say, $\\sim \\exp(S_N)$ at size $N$). We characterize all possible optimal matchings of a given instance of the problem, and we give a simple product formula for their number. Then, we study the probability distribution of $S_N$ (the zero-temperature entropy of the model), in the uniform random ensemble. We find that, for large $N$, $S_N \\sim \\frac{1}{2} N \\log N + N s + \\mathcal{O}\\left( \\log N \\right)$, where $s$ is a random variable whose distribution $p(s)$ does not depend on $N$. We give expressions for the asymptotics of the moments of $p(s)$, both from a formulation as a Brownian process, and via singularity analysis of the generating functions associated to $S_N$. The latter approach provides a combinatorial framework that allows to compute an asymptotic expansion to arbitrary order in $1/N$ for the mean and the variance of","url_abs":"https://arxiv.org/abs/2101.04926v1","url_pdf":"https://arxiv.org/pdf/2101.04926v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-number-of-optimal-matchings-for-euclidean","repo_url":"https://github.com/vittorioerba/EntropyMatching","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}