Extending Pólya's random walker beyond probability I. Complex weights

Working in combinatorial model ${\mathrm{W}}_{\mathrm{c}\mathrm{o}}(d)$, $d=1,2,\dots$, 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 ${\mathrm{W}}_{\mathrm{c}\mathrm{o}}(d)$ to standard models ${\mathrm{W}}_{\mathrm{M}\mathrm{a}}(d)$ based on Markov chains. The follow-up article will treat non-Archimedean models ${\mathrm{W}}_{\mathrm{f}\mathrm{o}}(k)$ in which weights are formal power series in $\mathbb{C}[[x_1,x_2,\dots ,x_k]]$.