  In the first part of this paper, we enumerate exactly walks on the square
lattice that start from the origin, but otherwise avoid the non positive
horizontal half-axis. We call them "walks on the slit plane". We count them by
their length, and by the coordinates of their endpoint. The corresponding three
variable generating function is algebraic of degree 8. Moreover, for any point
(i,j), the length generating function for walks of this type ending at (i,j) is
also algebraic, of degree 2 or 4, and involves the famous Catalan numbers. Our
method is based on the solution of a functional equation, established via a
simple combinatorial argument. It actually works for more general models, in
which walks take their steps in a finite subset of Z^2 satisfying two simple
conditions. The corresponding generating functions are always algebraic. In the
second part of the paper, we derive from our enumerative results a number of
probabilistic corollaries. For instance, we can compute exactly the probability
that an ordinary random walk starting from (i,j) hits for the first time the
horizontal half-axis at position (k,0), for any triple (i,j,k). This
generalizes a question raised by R. Kenyon, which was the starting point of
this paper. Taking uniformly at random all n-step walks on the slit plane, we
also compute the probability that they visit a given point (k,0), and the
average number of visits to this point. In other words, we quantify the
transience of the walks. Finally, we derive an explicit limit law for the
coordinates of their endpoint.
