Spanning trees in random regular uniform hypergraphs

Abstract

Let G n,r,s denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, . . . , n}. We analyse the asymptotic distribution of Y G , the number of spanning trees in G, when r, s 2 are fixed constants, (r, s) = (2, 2), and the necessary divisibility conditions hold. Greenhill, Kwan and Wind (2014) investigated the graph case (s = 2), providing an asymptotic formula for the expected value of Y G for any fixed r 3, which was previously only known up to a constant factor. They also found the asymptotic distribution of Y G for random cubic graphs (r = 3), and made a conjecture for arbitrary r 4. Here we prove this conjecture and extend the analysis to hypergraphs. When s 5 we prove a threshold result for the property that G n,r,s contains a spanning tree. We also calculate the asymptotic distribution of Y G for all fixed integers r, s 2 such that the probability that G n,r,s has a spanning tree tends to one as n grows.

The number of perfect matchings, and the nesting properties, of random regular graphs

“…The argument is then tuned to produce the distribution result of Z by Janson [14]. Recently, Greenhill, Isaev and Liang [11] proved that the number of spanning trees in (n, 𝑑) has the same type of distribution as (1). On the other hand, Garmo [10] studied the distributional phase transition of the number of 𝓁-cycles in (n, 𝑑) as 𝓁 grows from constant to linear in n. Its limiting distribution changes from a linear combination of independent Poisson variables to the exponential of that form, and the critical phase transition occurs when 𝓁 becomes linear in n.…”

The number of perfect matchings, and the nesting properties, of random regular graphs

“…The argument is then tuned to produce the distribution result of Z by Janson [13]. Recently, Greenhill, Isaev and Liang [10] proved that the number of spanning trees in G(n, d) has the same type of distribution as (1). On the other hand, Garmo [9] studied the distributional phase transition of the number of -cycles in G(n, d) as grows from constant to linear in n. Its limiting distribution changes from a linear combination of independent Poisson variables to the exponential of that form, and the critical phase transition occurs when becomes linear in n.…”