The asymptotic enhanced negative type of finite ultrametric spaces

Abstract: Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitative versions of these inequalities involving the so-called $p$-negative type gap. In particular, we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining. Let $(X,d)$ be a given finite ultrametric space with minimum non-zero distance $\alpha$. Then the $p$-negative type gap $\Gamma_{X}(p)$ of $(X,d)$ is positive for all $p \geq 0$. In this paper we compute the value of the limit \begin{eqnarray*} \Gamma_{X}(\infty) & = & \lim\limits_{p \rightarrow \infty} \frac{\Gamma_{X}(p)}{\alpha^{p}}. \end{eqnarray*} It turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula. On the basis of our calculations we are then able to characterize when $\Gamma_{X}(p)/ \alpha^{p}$ is constant on $[0, \infty)$. The determination of $\Gamma_{X}(\infty)$ also leads to new, asymptotically sharp, families of enhanced $p$-negative type inequalities for $(X,d)$. Indeed, suppose that $G \in (0, \Gamma_{X}(\infty))$. Then, for all sufficiently large $p$, we have \begin{eqnarray*} \frac{G \cdot \alpha^{p}}{2} \left( \sum\limits_{k=1}^{n} |\zeta_{k}| \right)^{2} + \sum\limits_{j,i =1}^{n} d(z_{j},z_{i})^{p} \zeta_{j} \zeta_{i} & \leq & 0 \end{eqnarray*} for each finite subset ${ z_{1}, \ldots, z_{n} } \subseteq X$ and each choice of real numbers $\zeta_{1}, \ldots, \zeta_{n}$ with $\zeta_{1} + \cdots + \zeta_{n} = 0$. We note that these results do not extend to general finite metric spaces.