Year: 2025
Type: preprint
Abstract: 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 $\mathbb{C}[[x_1,x_2,\dots,x_k]]$.
Source: arXiv (Cornell University)
Authors: Klazar, Martin, Horský, Richard
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417300486
Cites:
Cited by:
Related to: