  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.
