{"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/improved-replica-bounds-for-the-independence","title":"Improved replica bounds for the independence ratio of random regular graphs","arxiv_id":"2204.01353","date":"2022-04-04","proceeding":null,"authors":["Viktor Harangi"],"abstract":"Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $\\lambda$ tending to infinity. Finding the independence ratio of random $d$-regular graphs for some fixed degree $d$ has received much attention both in random graph theory and in statistical physics. For $d \\geq 20$ the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method shows that this 1-RSB formula is an upper bound for each $d \\geq 3$. For $d \\leq 19$ this bound is not tight and full-RSB is expected. In this work we use numerical optimization to find good substituting parameters for discrete $r$-RSB formulas ($r=2,3,4,5$) to obtain improved rigorous upper bounds for the independence ratio for each degree $3 \\leq d \\leq 19$. As $r$ grows, these formulas get increasingly complicated and it becomes challenging to compute their numerical values efficiently. Also, the functions to minimize have a large number of local minima, making global optimization a difficult task.","url_abs":"https://arxiv.org/abs/2204.01353v4","url_pdf":"https://arxiv.org/pdf/2204.01353v4.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":"improved-replica-bounds-for-the-independence","repo_url":"https://github.com/harangi/rsb","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}