Extending Pólya's random walker beyond probability I. Complex weights
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910972349775872 |
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| author | Klazar, Martin Horský, Richard |
| author_facet | Klazar, Martin Horský, Richard |
| contents | Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of Pólya's random walker in $\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a semi-formal approach to generating functions, we extend both theorems beyond probability to a more general model $\mathrm{W_{\mathbb{C}}}$ with complex weights. We relate models $\mathrm{W_{co}}(d)$ to standard models $\mathrm{W_{Ma}}(d)$ based on Markov chains. The follow-up article will treat non-Archimedean models $\mathrm{W_{fo}}(k)$ in which weights are formal power series in $\mathbb{C}[[x_1,x_2,\dots,x_k]]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending Pólya's random walker beyond probability I. Complex weights Klazar, Martin Horský, Richard Probability Combinatorics History and Overview 05A15 Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of Pólya's random walker in $\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a semi-formal approach to generating functions, we extend both theorems beyond probability to a more general model $\mathrm{W_{\mathbb{C}}}$ with complex weights. We relate models $\mathrm{W_{co}}(d)$ to standard models $\mathrm{W_{Ma}}(d)$ based on Markov chains. The follow-up article will treat non-Archimedean models $\mathrm{W_{fo}}(k)$ in which weights are formal power series in $\mathbb{C}[[x_1,x_2,\dots,x_k]]$. |
| title | Extending Pólya's random walker beyond probability I. Complex weights |
| topic | Probability Combinatorics History and Overview 05A15 |
| url | https://arxiv.org/abs/2505.12170 |