Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds

We show that for all positive, nonincreasing, convex functions $f:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ and for any collection of $n$ points $x_i\in \mathbb{R}$ with $x_1<x_2<\dots <x_n$ and $n$ even, the Pfaffian of the skew matrix defined by $f(x_j-x_i)$ above the diagonal is positive. More generally, we establish coefficientwise nonnegativity for Pfaffians arising from interval decompositions, which also yields quantitative lower bounds for singly and doubly bordered Pfaffians. Choosing $f(r)=r^{-2}$ proves the three nonvanishing assertions in the conjecture of Albouy and Moeckel. Further, we show that the rainbow bound (the product over the nested matching $\{1,n\},\{2,n-1\},\dots \{n\mathrm{/}2,n\mathrm{/}2+1\}$) is the unique single-matching product that bounds $\mathrm{P}\mathrm{f}(A_f(x))$ below uniformly over $x$ and $f$ positive, nonincreasing, convex.