Fixed forests in the minimum spanning tree and cubic volume growth

Abstract

Let $M_n$ be the minimum spanning tree of the complete graph $K_n$ with i.i.d.\ uniform edge weights. For a fixed forest $F$ with connected components $T_1, \ldots, T_d$, we show that there exists a function $\Psi$ on finite trees such that $$ n^{|E(F)|} \mathbb{P}_n(F \subseteq M_n) \longrightarrow \prod_{i=1}^d \Psi(T_i). $$ We give a recursive description of $\Psi$ and calculate it explicitly for several small trees. For the star $S_k$ and the path $P_k$, we prove that $\Psi(S_k) \sim \zeta(2)^k$ and $\Psi(P_k) \sim k^2/12$, respectively. We also show that the expected size of a ball of radius $r$ is asymptotic to $r^3/36$, and give exponential tail bounds.