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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2212.01702 |
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| _version_ | 1866910656972718080 |
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| author | Dumitraşcu, Dorin Suciu, Liviu |
| author_facet | Dumitraşcu, Dorin Suciu, Liviu |
| contents | We give precise asymptotics to the number of first time returning random walks in the standard orthogonal lattice in $\mathbb{R}$ and we prove that these numbers do not form a $P$-recursive sequence. In the process, the known asymptotics of the number of closed walks are obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. By showing that the relevant generating functions are $G$-functions, we use a form of the Hadamard convolution to find their singularities in all dimensions and give the ODEs that they satisfy for $d\leq 5$, some of which seem to be new. We use the Lucas property of the number of closed walks to prove that the corresponding generating function is not invertible as a $G$-function, which immediately implies that the generating function of the first time returning walks is not holonomic. We propose a few conjectures on the form of the asymptotic coefficients and of the ODEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01702 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Asymptotics and the Non-Holonomic Character of First Returns in the Standard Euclidean Lattice Dumitraşcu, Dorin Suciu, Liviu Combinatorics Probability 05A16 (Primary) 05A10, 40E99, 33C45 (Secondary) We give precise asymptotics to the number of first time returning random walks in the standard orthogonal lattice in $\mathbb{R}$ and we prove that these numbers do not form a $P$-recursive sequence. In the process, the known asymptotics of the number of closed walks are obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. By showing that the relevant generating functions are $G$-functions, we use a form of the Hadamard convolution to find their singularities in all dimensions and give the ODEs that they satisfy for $d\leq 5$, some of which seem to be new. We use the Lucas property of the number of closed walks to prove that the corresponding generating function is not invertible as a $G$-function, which immediately implies that the generating function of the first time returning walks is not holonomic. We propose a few conjectures on the form of the asymptotic coefficients and of the ODEs. |
| title | On the Asymptotics and the Non-Holonomic Character of First Returns in the Standard Euclidean Lattice |
| topic | Combinatorics Probability 05A16 (Primary) 05A10, 40E99, 33C45 (Secondary) |
| url | https://arxiv.org/abs/2212.01702 |