Barrier Log-Concavity of Large Forests

Source abstract

We prove that the independence sequence of every sufficiently large forest is strictly barrier log-concave: whenever $i_{k+1}>i_k$, one has $i_k^2>i_{k-1}i_{k+1}$. Consequently, every common downward coefficient tilt is unimodal, with one order threshold valid simultaneously for all $0<t\le1$. 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 $\lfloor17α/25\rfloor$; 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.