in several variables [@PeWi13].

Coupling these techniques – diagonal representation and analytic combinatorics in several variables – yields explicit, yet simple, asymptotic formulas for families of lattice paths. The focus of this article is $d$-dimensional models whose set of allowable steps is symmetric with respect to any axis; we call these models *highly symmetric* walks. The techniques of analytic combinatorics in several variables apply in a rather straightforward way to derive dominant asymptotics for the number of walks ending anywhere and give an effective procedure to calculate descending terms in the asymptotic expansions. Furthermore, we also consider the subfamily of walks that return to the origin (known as *excursions*). Once our equations are established, they are suitable input to existing implementations such as that of Raichev [@Raic12] (however, in practice one can calculate only the first few terms in these expansions).

The highly symmetric walks we present are amenable to a kernel method treatment. In particular, they fit well into the ongoing study of lattice path classes restricted to an orthant and taking only “small” steps [@BoMi10; @BoRaSa14]. This collection of models forms a little universe exhibiting many interesting phenomena, and recent work in two and three dimensions has used 