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 $0<t\le1$. 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 $Θ=\tfrac12e^{3-2W(1/e)}$ 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.