Partially directed paths in a symmetric wedge

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and its solution. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to a symmetric wedge defined by Y = ±pX, where p > 0 is an integer. We prove that the growth constant for all these models is equal to 1+ √ 2, independent of the angle of the wedge. We derive a functional equation for the generating function of the model, and obtain an explicit solution when p = 1. From this we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p = 1. The functional equation is solved by a variation of the kernel method, which we call the “iterated kernel method”. This method appears to be similar to the obstinate kernel method used by Bousquet-Mélou (see, for example [Bousqet-Mélou, Electronic J. Combinatorics, 9 (2003), R19]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating function turns out to be similar in form to Jacobi θ-functions, and has a natural boundary on the unit circle. Abstract in french goes here.in french goes here.