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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.16688 |
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| _version_ | 1866913134414921728 |
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| author | Mishna, Marni Pulido, Juan |
| author_facet | Mishna, Marni Pulido, Juan |
| contents | We report on the status of the conjecture of Bousquet-Mélou and Mishna that the univariate counting generating function of a small-step quarter-plane lattice model is D-finite if and only if the group of the walk is finite. While the finite-group case is fully resolved, the infinite-group case remains incomplete. We list the arguments for the non-D-finiteness for 21 of the 56 infinite-group models: the five singular models, three models with zero drift and thirteen models with polar interior drift. The proof of the latter two families uses asymptotic results of Bostan--Raschel--Salvy combined with probabilistic estimates of Denisov--Wachtel and Duraj. We further identify nine infinite-group models whose endpoint counting series are differentially algebraic via decoupling functions, though this does not settle their D-finiteness. For 21 of the remaining models, numerical estimation of singular exponents suggests non-D-finiteness of one of its the boundary series $Q(1,0;t)$ or $Q(0,1;t)$, and we state a conjecture to this effect. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16688 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the small-step quarter plane lattice walks with a non D-finite univariate generating function Mishna, Marni Pulido, Juan Combinatorics Probability 05A15 We report on the status of the conjecture of Bousquet-Mélou and Mishna that the univariate counting generating function of a small-step quarter-plane lattice model is D-finite if and only if the group of the walk is finite. While the finite-group case is fully resolved, the infinite-group case remains incomplete. We list the arguments for the non-D-finiteness for 21 of the 56 infinite-group models: the five singular models, three models with zero drift and thirteen models with polar interior drift. The proof of the latter two families uses asymptotic results of Bostan--Raschel--Salvy combined with probabilistic estimates of Denisov--Wachtel and Duraj. We further identify nine infinite-group models whose endpoint counting series are differentially algebraic via decoupling functions, though this does not settle their D-finiteness. For 21 of the remaining models, numerical estimation of singular exponents suggests non-D-finiteness of one of its the boundary series $Q(1,0;t)$ or $Q(0,1;t)$, and we state a conjecture to this effect. |
| title | On the small-step quarter plane lattice walks with a non D-finite univariate generating function |
| topic | Combinatorics Probability 05A15 |
| url | https://arxiv.org/abs/2605.16688 |