Barrier Log-Concavity of Large Forests
Abstract
We prove that the independence sequence of every sufficiently large forest is strictly barrier log-concave: whenever ik+1>ik, one has ik2>ik−1ik+1. Consequently, every common downward coefficient tilt is unimodal, with one order threshold valid simultaneously for all 0<t≤1. As an application, we prove that every sufficiently large well-covered tree has a strictly log-concave independence polynomial. More generally, the latter conclusion holds for well-covered forests with sufficiently large matching number. Every sufficiently large forest with a palindromic independence polynomial is also strictly log-concave, with its modes at the central ranks. The proof establishes local coefficient-curvature estimates uniformly over forests of unbounded maximum degree and over activities tending to zero. Combined with the fixed-density theorem of Fang, Lu, Nevo, Yao, and Zheng, these estimates complete the strict log-concavity interval down to rank one through ⌊17α/25⌋; the terminal fraction is inherited. The classical decreasing tail then yields the barrier property. A cumulative-coefficient corona theorem transfers barrier log-concavity of the base forest to full strict log-concavity of its leaf corona.
Related
- Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems
- Unimodality of Independence Polynomials for Sufficiently Large Forests
- Quagmires and large Suslin forests
- Uniform Large Deviations, Barrier Identity and the Néel--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation
- A counterexample to a log-concavity conjecture of Brenti
BibTeX
@misc{levit2026barrier,
title = {{Barrier Log-Concavity of Large Forests}},
author = {Vadim E. Levit},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22098},
eprint = {2610.22098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22098},
}
RIS
TY - UNPB
TI - Barrier Log-Concavity of Large Forests
AU - Vadim E. Levit
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22098
UR - https://arxiv.org/abs/2610.22098
AB - We prove that the independence sequence of every sufficiently large forest is strictly barrier log-concave: whenever i_k+1>i_k, one has i_k^2>i_k−1i_k+1. Consequently, every common downward coefficient tilt is unimodal, with one order threshold valid simultaneously for all 0<t≤1. As an application, we prove that every sufficiently large well-covered tree has a strictly log-concave independence polynomial. More generally, the latter conclusion holds for well-covered forests with sufficiently large matching number. Every sufficiently large forest with a palindromic independence polynomial is also strictly log-concave, with its modes at the central ranks. The proof establishes local coefficient-curvature estimates uniformly over forests of unbounded maximum degree and over activities tending to zero. Combined with the fixed-density theorem of Fang, Lu, Nevo, Yao, and Zheng, these estimates complete the strict log-concavity interval down to rank one through ⌊17α/25⌋; the terminal fraction is inherited. The classical decreasing tail then yields the barrier property. A cumulative-coefficient corona theorem transfers barrier log-concavity of the base forest to full strict log-concavity of its leaf corona.
ER -