Extending P\'olya's random walker beyond probability I. Complex weights

TL;DR

This paper extends classical Pólya's random walk models to complex weights, proving recurrence theorems in higher dimensions and relating combinatorial models to Markov chains, with implications for non-Archimedean models.

Contribution

It generalizes Pólya's random walk to complex weights and establishes recurrence results, connecting combinatorial and Markov chain models.

Findings

Recurrence theorems for complex-weighted models

Effective recurrence criteria in 2D

Relation between combinatorial and Markov chain models

Abstract

Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of P\'olya'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 $\mathbb{C}[[x_1,x_2,\dots,x_k]]$.