Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds
Abstract
We show that for all positive, nonincreasing, convex functions f : (0, ∞) → (0,∞) and for any collection of n points xi ∈ ℝ with x1 < x2 < … < xn and n even, the Pfaffian of the skew matrix defined by f(xj − xi) 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}, … {n/2, n/2 + 1}) is the unique single-matching product that bounds Pf(Af(x)) below uniformly over x and f positive, nonincreasing, convex.
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BibTeX
@misc{johnson2026pfaffian,
title = {{Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds}},
author = {William Johnson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24371},
eprint = {2610.24371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24371},
}
RIS
TY - UNPB
TI - Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds
AU - William Johnson
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24371
UR - https://arxiv.org/abs/2610.24371
AB - We show that for all positive, nonincreasing, convex functions f : (0, ∞) → (0,∞) and for any collection of n points x_i ∈ ℝ with x_1 < x_2 < … < 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}, … {n/2, n/2 + 1}) is the unique single-matching product that bounds Pf(A_f(x)) below uniformly over x and f positive, nonincreasing, convex.
ER -