Representation theory and cycle statistics for random walks on the symmetric group

[2512.13969] Representation theory and cycle statistics for random walks on the symmetric group Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2512.13969 (math) [Submitted on 16 Dec 2025 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Representation theory and cycle statistics for random walks on the symmetric group Authors: Dominic Arcona View PDF HTML (experimental) Abstract: We use representation theory of $S_n$ to analyze the mixing of cycle type statistics $a_j(\sigma) = \{\text{# of $j$-cycles of $\sigma$}\}$ for any fixed $j$ in permutations $\sigma_t$ resulting from the $t$-step random transposition walk on $S_n$. We also derive analogous results for the star transposition walk. Our approach uses the method of moments; a key ingredient is a new formula for the coefficients in the irreducible character decomposition of the $S_n$-class function $(a_j)^r(\sigma)=\{(\text{# of $j$-cycles of $\sigma$})^r\}$ for any positive integers $r,j$ when $n\geq 2rj$. Comments: An update to a previous version of this paper which appeared December 2025 including a corrected result concerning the random transposition walk. 47 pages, 4 figures. Comments welcome! Subjects: Combinatorics (math.CO) ; Probability (math.PR); Representation Theory (math.RT) Cite as: arXiv:2512.13969 [math.CO] (or arXiv:2512.13969v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2512.13969 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Dominic Arcona [ view email ] [v1] Tue, 16 Dec 2025 00:16:21 UTC (33 KB) [v2] Thu, 27 Aug 2026 20:31:50 UTC (51 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2025-12 Change to browse by: math math.PR math.RT References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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