The asymptotic enhanced negative type of finite ultrametric spaces

TL;DR

This paper investigates the asymptotic behavior of the negative type gap in finite ultrametric spaces, providing explicit formulas and inequalities that enhance understanding of their embedding properties.

Contribution

It computes the limit of the scaled negative type gap as p approaches infinity for finite ultrametric spaces and characterizes when this ratio remains constant.

Findings

Explicit formula for the limit of the negative type gap as p approaches infinity.

Characterization of when the scaled negative type gap is constant.

New asymptotically sharp inequalities for 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.