  A spanning tree $T$ in a graph $G$ is a sub-graph of $G$ with the same vertex
set as $G$ which is a tree. In 1981, McKay proved an asymptotic result
regarding the number of spanning trees in random $k$-regular graphs. In this
paper we prove a high-dimensional generalization of McKay's result for random
$d$-dimensional, $k$-regular simplicial complexes on $n$ vertices, showing that
the weighted number of simplicial spanning trees is of order
$(\xi_{d,k}+o(1))^{\binom{n}{d}}$ as $n\to\infty$, where $\xi_{d,k}$ is an
explicit constant, provided $k> 4d^2+d+2$. A key ingredient in our proof is the
local convergence of such random complexes to the $d$-dimensional, $k$-regular
arboreal complex, which allows us to generalize McKay's result regarding the
Kesten-McKay distribution.
