  We consider planar lattice walks that start from (0,0), remain inthe first
quadrant i, j >= 0, and are made of three types of steps: North-East, West and
South. These walks are known to have remarkable enumerative and probabilistic
properties: -- they are counted by nice numbers (Kreweras 1965), -- the
generating function of these numbers is algebraic (Gessel 1986), -- the
stationary distribution of the corresponding Markov chain in the quadrant has
an algebraic probability generating function (Flatto and Hahn 1984). These
results are not well understood, and have been established via complicated
proofs. Here we give a uniform derivation of all of them, whichis more
elementary that those previously published.We then go further by computing the
full law of the Markov chain. This helps to delimit the border of algebraicity:
the associated probability generating function is no longer algebraic, unless a
diagonal symmetry holds. Our proofs are based on the solution of certain
functional equations,which are very simple to establish. Finding purely
combinatorial proofs remains an open problem.
