2020/08/16 by Rosenthal, Ron, Tenenbaum, Lior
#05E45 #60B10 #60B20 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2008.06955
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 (ξd,k+o(1))^\binomnd as n→∞, where ξd,k is an explicit constant, provided k> 4d2+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.