Balanced weighted Motzkin paths: Pearson structure and saddlepoint asymptotics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918303768772608 |
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| author | Omelchenko, Alexander |
| author_facet | Omelchenko, Alexander |
| contents | We analyse weighted Motzkin paths with step multiplicities that vary linearly with height. In the balanced case the associated exponential generating function satisfies a Pearson-type PDE, and solving by characteristics yields closed expressions in all drift regimes. These formulas reveal a moving algebraic singularity that governs both local and global behaviour. Locally this gives a Gaussian central window for the terminal-height distribution, while globally we identify an explicit limit cumulant generating function and prove an $n$-speed large-deviation principle. For finite $n$, Daniels' lattice saddlepoint approximation provides a single formula that is accurate across the full range of $k$; in all quadratic regimes it achieves a uniform interior relative error of order $n^{-1}$. The results link Pearson geometry with uniform saddlepoint methods and extend naturally to other weighted path models and tridiagonal recurrences. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Balanced weighted Motzkin paths: Pearson structure and saddlepoint asymptotics Omelchenko, Alexander Probability Combinatorics 05A16, 60F10, 60C05 G.2.1 We analyse weighted Motzkin paths with step multiplicities that vary linearly with height. In the balanced case the associated exponential generating function satisfies a Pearson-type PDE, and solving by characteristics yields closed expressions in all drift regimes. These formulas reveal a moving algebraic singularity that governs both local and global behaviour. Locally this gives a Gaussian central window for the terminal-height distribution, while globally we identify an explicit limit cumulant generating function and prove an $n$-speed large-deviation principle. For finite $n$, Daniels' lattice saddlepoint approximation provides a single formula that is accurate across the full range of $k$; in all quadratic regimes it achieves a uniform interior relative error of order $n^{-1}$. The results link Pearson geometry with uniform saddlepoint methods and extend naturally to other weighted path models and tridiagonal recurrences. |
| title | Balanced weighted Motzkin paths: Pearson structure and saddlepoint asymptotics |
| topic | Probability Combinatorics 05A16, 60F10, 60C05 G.2.1 |
| url | https://arxiv.org/abs/2601.17634 |