coordinate (this implies that for each coordinate there is a walk in the model which moves in that coordinate).

The number of walks taking steps in ${\ensuremath{\mathcal{S}}}$ which are restricted to the positive orthant $\mathbb{N}^d = \mathbb{Z}_{\geq0}^d$ are studied by expressing the counting generating functions of such models as positive parts of multivariate rational Laurent series, which are then converted to diagonals of rational functions in $d+1$ variables. A first consequence is that all of these models have D-finite generating functions (since D-finite functions are closed under the diagonal operation).

After the above manipulations, these models are very well suited to the asymptotic enumeration methods for diagonals of rational functions outlined in [@PeWi13], in particular the cases which were developed by Pemantle, Raichev and Wilson in [@PeWi02] and [@RaWi08]. Following these methods, we study the singular variety of the denominator of this rational function to determine related asymptotics. The condition of having a symmetry across each axis ensures that the variety is smooth and allows us to calculate the leading asymptotic term explicitly. This is not generally the case, in our experience, and hence we focus on this particular kind of restriction.

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