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A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials
Abstract
Let πn be a uniform parking function of size n. We determine the scale at which its first k coordinates cease to be close in total variation to k independent uniform variables on {1,…,n}. The distance is comparable to 1∧ √(klog(k+1)/n), uniformly for n≥2 and 1≤ k≤ n, and tends to one above the scale n/log n. At the critical scale k∼λn/log n, it converges to 2Φ(√(λ/π))−1, where Φ is the standard normal distribution function. A key ingredient is the Lorentzian property of the occupation generating polynomial of parking functions. Together with a strengthened Han inequality, this yields a relative entropy estimate for approximation by the product of the one-coordinate marginals.
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BibTeX
@misc{kortchemski2026total,
title = {{A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials}},
author = {Igor Kortchemski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40082},
eprint = {2610.40082},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40082},
}
RIS
TY - UNPB
TI - A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials
AU - Igor Kortchemski
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40082
UR - https://arxiv.org/abs/2610.40082
AB - Let π_n be a uniform parking function of size n. We determine the scale at which its first k coordinates cease to be close in total variation to k independent uniform variables on {1,…,n}. The distance is comparable to 1∧ √(klog(k+1)/n), uniformly for n≥2 and 1≤ k≤ n, and tends to one above the scale n/log n. At the critical scale k∼λn/log n, it converges to 2Φ(√(λ/π))−1, where Φ is the standard normal distribution function. A key ingredient is the Lorentzian property of the occupation generating polynomial of parking functions. Together with a strengthened Han inequality, this yields a relative entropy estimate for approximation by the product of the one-coordinate marginals.
ER -