  Two-dimensional (random) walks in cones are very natural both in
combinatorics and probability theory: they are interesting for themselves and
also because they are strongly related to other discrete structures. While
walks restricted to the first quadrant have been studied a lot, the case of
planar, non-convex cones---equivalent to the three-quarter plane after a linear
transform---has been approached only recently. In this article we develop an
analytic approach to the case of walks in three quadrants. The advantage of
this method is to provide uniform treatment in the study of models
corresponding to different step sets. After splitting the three quadrants in
two symmetric convex cones, the method is composed of three main steps: write a
system of functional equations satisfied by the counting generating function,
which may be simplified into one single equation under symmetry conditions;
transform the functional equation into a boundary value problem; and finally
solve this problem, using a concept of anti-Tutte's invariant. The result is a
contour-integral expression for the generating function. Such systems of
functional equations also appear in queueing theory with the famous
Join-the-Shortest-Queue model, which is still an open problem in the
non-symmetric case.
