Simplicial Spanning Trees in Random Steiner Complexes

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 (?(d,k) + o(1))(n/d) as n? 8, where ?(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.