Asymptotic negative type properties of finite ultrametric spaces

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)(X,d) be a given finite ultrametric space with minimum non-zero distance α. Then the p  -negative type gap ΓX(p)ΓX(p) of (X,d)(X,d) is positive for all p≥0p≥0. In this paper we compute the value of the limitΓX(∞):=limp→∞⁡ΓX(p)αp. It turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula. This formula allows us to characterize when the ratio ΓX(p)/αpΓX(p)/αp is a constant independent of p  . The determination of ΓX(∞)ΓX(∞) also leads to new, asymptotically sharp, families of enhanced p  -negative type inequalities for (X,d)(X,d). Indeed, suppose that G∈(0,ΓX(∞))G∈(0,ΓX(∞)). Then, for all sufficiently large p, the inequalityG⋅αp2(∑k=1n|ζk|)2+∑j,i=1nd(zj,zi)pζjζi≤0 holds for each finite subset {z1,…,zn}⊆X{z1,…,zn}⊆X, and each scalar n  -tuple ζ=(ζ1,…,ζn)∈Rnζ=(ζ1,…,ζn)∈Rn that satisfies ζ1+⋯+ζn=0ζ1+⋯+ζn=0. Notably, these results do not extend to general finite metric spaces.