  We propose a new approach for finding discrete harmonic functions in the
quarter plane with Dirichlet conditions. It is based on solving functional
equations that are satisfied by the generating functions of the values taken by
the harmonic functions. As a first application of our results, we obtain a
simple expression for the harmonic function that governs the asymptotic tail
distribution of the first exit time for random walks from the quarter plane. As
another corollary, we prove, in the zero drift case, the uniqueness of the
discrete harmonic function.
