Counting paths on the slit plane
Identifieur interne : 001A19 ( Istex/Curation ); précédent : 001A18; suivant : 001A20Counting paths on the slit plane
Auteurs : Mireille Bousquet-Mélou [France] ; Gilles Schaeffer [France]Source :
- Trends in Mathematics
Abstract
Abstract: 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 $$ \sqrt {n}$$ (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 first time the horizontal half-line
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DOI: 10.1007/978-3-0348-8405-1_9
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