Walks in the slit plane
Résumé
In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line ${\cal H}=\{(k,0), k\le 0\}$. We call them {\em walks on the slit plane\/}. We count them by their length, and by the coordinates of their endpoint. The corresponding three variable \gf \ is algebraic of degree $8$. Moreover, for any point $(i,j)$, the length \gf \ 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 $\zs ^2$ satisfying two simple conditions. The corresponding \gfs \ 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 half-line $\cal H$ 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.