Plane bipolar orientations and quadrant walks

TL;DR

This paper studies refined enumeration of plane bipolar orientations with face degree constraints, connecting them to quadrant lattice walks, and provides D-finite generating functions and asymptotic estimates using combinatorial and probabilistic methods.

Contribution

It introduces a new enumeration approach for bipolar orientations with face degree restrictions via lattice walk bijections and proves their generating functions are D-finite.

Findings

Generating functions are D-finite when face degrees are bounded.

Established asymptotic estimates for enumeration numbers.

Connected bipolar orientations to quadrant lattice walks, enabling new analysis.

Abstract

Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with $n$ edges are known to be counted by the $n$th Baxter number $b(n)$, which can be defined by a linear recurrence relation with polynomial coefficients. Equivalently, the associated generating function $\sum_n b(n)t^n$ is D-finite. In this paper, we address a much refined enumeration problem, where we record for every $r$ the number of faces of degree $r$. When these degrees are bounded, we show that the associated generating function is given as the constant term of a multivariate rational series, and thus is still D-finite. We also provide detailed asymptotic estimates for the corresponding numbers. The methods used earlier to count all plane bipolar orientations, regardless of their face degrees, do not generalize easily to record face degrees. Instead, we start from a recent bijection, due to Kenyon et al., that sends bipolar orientations onto certain lattice walks confined to the first quadrant. Due to this bijection, the study of bipolar orientations meets the study of walks confined to a cone, which has been extremely active in the past 15 years. Some of our proofs rely on recent developments in this field, while others are purely bijective. Our asymptotic results also involve probabilistic arguments.