Counting paths on the slit plane

We present a method, based on functional equations, to enumerate paths on the square lattice that avoid a horizontal half-line. The corresponding generating functions are algebraic, and sometimes remarkably simple: for instance, the number of paths of length 2n + 1 going from (0; 0) to (1; 0) and avoiding the nonpositive horizontal axis (except at their starting point) is C2n+1 , the (2n + 1)th Catalan number. More generally, we enumerate exactly all paths of length n starting from (0; 0) and avoiding the nonpositive horizontal axis. We then obtain limit laws for the coordinates of their endpoint: in particular, the average abscissa of their endpoint grows like p n (up to an explicit multiplicative constant), which shows that these paths are strongly repelled from the origin. We derive from our results the distribution of the position where a random walk, starting from a given point, hits for the rst time the horizontal half-line.