highly symmetric walks as diagonals of rational functions. We then perform an asymptotic analysis of the coefficients of counting generating functions using techniques from the study of analytic combinatorics in several variables, and consequently link some of the combinatorial symmetries in a walk model to both analytic properties of the generating function and geometric properties of an associated variety. After this is complete, we examine how this connects to the notion of walks in Weyl chambers, use results from their study to determine asymptotic results about excursions, and discuss how the Weyl chamber viewpoint can be used in future work to examine larger classes of lattice path models through diagonals. Next we specify the walks we study in order to precisely state our main results.

Highly Symmetric Walks
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Concretely, the lattice path models we consider are restricted as follows. For a fixed dimension $d$, we define a model by its step set ${\ensuremath{\mathcal{S}}}\subseteq \{\pm1,0\}^d \setminus \{\mathbf{0}\}$ and say that ${\ensuremath{\mathcal{S}}}$ is *symmetric about the $x_k$ axis* if $(i_1,\dots,i_k,\dots,i_d) \in {\ensuremath{\mathcal{S}}}$ implies $(i_1,\dots,-i_k,\dots,i_d) \in {\ensuremath{\mathcal{S}}}$. We further impose a non-triviality condition: for each coordinate there is at least one step in ${\ensuremath{\mathcal{S}}}$ which moves in the positive direction of that 