Variance Bounds and Strict Log-Concavity for Forest Coronas
Abstract
We prove that the size of a random independent set in every nonempty forest has variance strictly smaller than its mean at common activity 0<t≤1. A rooted quadratic reserve gives quantitative forms of this inequality. Mean matching transfers the bounds to fixed-weight canonical completion laws, and exact binomial moments prove strict log-concavity for every forest corona and every well-covered forest. The stronger factorial normalization ((m−j)!ij)j=0m is strictly log-concave; we give explicit coefficient margins, terminal sharpness, one-less smoothing, and the exact universal three-index binomial ULC prefix. An exact attachment argument doubles the rooted reserve for every nontrivial unweighted tree. For a single marked root, a square reserve holds through Θ=1/2e3−2W(1/e) and for every positive mark at roots of degree at most four. A stronger reserve holds through mark 15/14. These estimates yield marked clique, adjacent-pair, and repulsive two-piece variance bounds with fixed-weight canonical transfer. Finally, arbitrary positive site activities in [3t/4,t] satisfy a uniform rooted reserve on every forest. This proves strict log-concavity for weighted forest coronas with compatible base marks. An explicit weighted-star obstruction shows why an unrestricted heterogeneous extension is impossible. Complete proofs are included, together with the required classical graph characterization and variance comparison.
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BibTeX
@misc{levit2026variance,
title = {{Variance Bounds and Strict Log-Concavity for Forest Coronas}},
author = {Vadim E. Levit},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41102},
eprint = {2610.41102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41102},
}
RIS
TY - UNPB
TI - Variance Bounds and Strict Log-Concavity for Forest Coronas
AU - Vadim E. Levit
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41102
UR - https://arxiv.org/abs/2610.41102
AB - We prove that the size of a random independent set in every nonempty forest has variance strictly smaller than its mean at common activity 0<t≤1. A rooted quadratic reserve gives quantitative forms of this inequality. Mean matching transfers the bounds to fixed-weight canonical completion laws, and exact binomial moments prove strict log-concavity for every forest corona and every well-covered forest. The stronger factorial normalization ((m−j)!i_j)_j=0^m is strictly log-concave; we give explicit coefficient margins, terminal sharpness, one-less smoothing, and the exact universal three-index binomial ULC prefix. An exact attachment argument doubles the rooted reserve for every nontrivial unweighted tree. For a single marked root, a square reserve holds through Θ=1/2e^3−2W(1/e) and for every positive mark at roots of degree at most four. A stronger reserve holds through mark 15/14. These estimates yield marked clique, adjacent-pair, and repulsive two-piece variance bounds with fixed-weight canonical transfer. Finally, arbitrary positive site activities in [3t/4,t] satisfy a uniform rooted reserve on every forest. This proves strict log-concavity for weighted forest coronas with compatible base marks. An explicit weighted-star obstruction shows why an unrestricted heterogeneous extension is impossible. Complete proofs are included, together with the required classical graph characterization and variance comparison.
ER -