Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks

[2212.09919] Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2212.09919 (math) [Submitted on 19 Dec 2022 ( v1 ), last revised 24 Aug 2026 (this version, v2)] Title: Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks Authors: Cetin Hakimoglu-Brown View PDF HTML (experimental) Abstract: We consider the two-barrier walk on the integers with steps $+y$ and $-b$, stopped on entering either absorbing set. Its finite-state generating functions are rational and admit the standard Schur-function representation in the roots of the normalized kernel $P_z(X)=X^{b+y}-2X^b+z$. Our contribution is a two-stage pruning method for constructing these root expansions without computing terms that cannot affect the final polynomial. First, the Weyl expansion is grouped into complete small-root/large-root blocks, and a $z$-adic valuation bound removes every block beginning above the denominator degree. Second, surviving blocks are split into residue subblocks and projected onto complete cyclic root-label orbits; the resulting congruences eliminate nontrivial characters without any termwise rationality test. In the $(+3,-2)$, $m=12$ example only one mixed-block degree survives, while for $(+5,-2)$, $m=22$, twenty of twenty-one Weyl terms are excluded before expansion. Comments: 15 pages Subjects: Probability (math.PR) ; Combinatorics (math.CO) ACM classes: G.3; G.2; G.2.1, G.2.m Cite as: arXiv:2212.09919 [math.PR] (or arXiv:2212.09919v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2212.09919 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Cetin Hakimoglu [ view email ] [v1] Mon, 19 Dec 2022 23:58:02 UTC (18 KB) [v2] Mon, 24 Aug 2026 21:56:22 UTC (16 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2022-12 Change to browse by: math math.CO References & Citations SciX Google Scholar Semantic Scholar 1 blog link ( what is this? ) 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)

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