Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds

Abstract

We show that for all positive, nonincreasing, convex functions $f : (0, \infty) \to (0,\infty)$ and for any collection of $n$ points $x_i \in \mathbb{R}$ with $x_1 < x_2 < \ldots < x_n$ 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 $\{1,n\}, \{2,n-1\}, \ldots \{n/2, n/2 + 1\}$) is the unique single-matching product that bounds $\mathrm{Pf}(A_f(x))$ below uniformly over $x$ and $f$ positive, nonincreasing, convex.