Extending Pólya's random walker beyond probability I. Complex weights
Martin Klazar, Richard Horský
TL;DR
This work generalizes Pólya’s random walker beyond probability by introducing combinatorial and complex-weight models $W_{co}(d)$ and $W_{\\mathbb{C}}$, and relating them to standard Markov-walker models $W_{Ma}(d)$. A generating-function framework is developed to derive strengthened recurrence results, yielding explicit limits for recurrence probabilities in terms of $B(1)$ and $B_0(1)$, and, in two dimensions, providing effective, computable bounds. The authors extend the analysis to complex weights using a semi-formal approach to generating functions (SFA), establishing formal relations among $A_h,B_h,C_h,D_h$ and proving recurrence limits under broad convergence hypotheses. Abel-type results underpin the translation from formal series to actual limits, and the paper sets the stage for non-Archimedean and formal power-series weight models in a follow-up work. Overall, the paper broadens the scope of Pólya-type recurrence phenomena to richer weight schemes while preserving computable asymptotics and structural connections to Markov-walker theory.
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 $\mathbb{C}[[x_1,x_2,\dots,x_k]]$.