Christoffel transformations of symmetric d-connections and gap probability recurrences
Abstract
We study the Christoffel transformation w(x)↦ x2w(x) of an even discrete weight. For the associated orthogonal polynomial ensembles on squared nodes y=x2, ratios of moment determinants relate the largest-particle distributions and transfer exact gap probability recurrences between the ensembles. For positive weights satisfying the stated moment and support assumptions, this comparison requires no Pearson condition. The transformation is realized by elementary bundle modifications at zero and infinity on ℙ1. When the original weight has Pearson ratio $d(-z-1)/d(z)$ with deg d=1,2, these modifications conjugate cutoff dynamics on common stable loci of symmetric d-connections under the stated geometric hypotheses. The transformed recurrence coefficients are expressed explicitly as rational functions of spectral coordinates. On the unit lattice, a local recurrence formula applies to rational symmetric Pearson ratios of arbitrary degree, including outside these charts, whenever its local hypotheses hold.
Related
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- The capillary Lp dual Christoffel-Minkowski problem for 1<p<q≤ k+1 with 1≤ k≤ n
BibTeX
@misc{selemenchuk2026christoffel,
title = {{Christoffel transformations of symmetric $d$-connections and gap probability recurrences}},
author = {Anton Selemenchuk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28497},
eprint = {2610.28497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28497},
}
RIS
TY - UNPB
TI - Christoffel transformations of symmetric d-connections and gap probability recurrences
AU - Anton Selemenchuk
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.28497
UR - https://arxiv.org/abs/2610.28497
AB - We study the Christoffel transformation w(x)↦ x^2w(x) of an even discrete weight. For the associated orthogonal polynomial ensembles on squared nodes y=x^2, ratios of moment determinants relate the largest-particle distributions and transfer exact gap probability recurrences between the ensembles. For positive weights satisfying the stated moment and support assumptions, this comparison requires no Pearson condition. The transformation is realized by elementary bundle modifications at zero and infinity on ℙ^1. When the original weight has Pearson ratio $d(-z-1)/d(z)$ with deg d=1,2, these modifications conjugate cutoff dynamics on common stable loci of symmetric d-connections under the stated geometric hypotheses. The transformed recurrence coefficients are expressed explicitly as rational functions of spectral coordinates. On the unit lattice, a local recurrence formula applies to rational symmetric Pearson ratios of arbitrary degree, including outside these charts, whenever its local hypotheses hold.
ER -