Semiformal method I. Pólya's theorem for the complete graph $K_{\mathbb{N}}$ with complex edge weights
M. Klazar, R. Horský
TL;DR
The paper generalizes Pólya's visitation theorem from grid walks to walks on the countable complete graph $K_{\mathbb{N}}$ with complex edge weights, using a semiformal generating-function framework. It defines weights $h:\mathbb{N}_2\to\mathbb{C}$ and generating functions $A_{h,v}(x)$, and develops Abel-type limit transfer tools to handle both absolute and conditional convergence. The authors prove fourteen theorems covering the cases $v=1$ and $v\neq 1$, for convex and non-convex weights, including $v$-transitive structures, and obtain explicit asymptotic formulas (often involving square roots) for the sums $A_{h,v}(1)$ and limits $\lim_{n\to\infty} h(W(v,n))$, thereby extending Pólya's theorem to a complex-weight, countable-graph setting. The work introduces the semiformal method as a robust extension of the symbolic method to countable sets and signals future development for edge-weights in multivariate formal power series, with potential impact on enumerative combinatorics and probabilistic models on infinite graphs.
Abstract
Using the semiformal method in combinatorics we generalize Pólya's theorem. This theorem determines the limiting probability that walks in the grid graph $\mathbb{Z}^d$ visit the given vertex. We generalize it to the countable complete graph $K_{\mathbb{N}}$ with edge weights in $\mathbb{C}$. In part II we treat edge weights in $\mathbb{C}[[x_1,\dots,x_k]]$.