Degree laws and edge correlations for random forests
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Abstract
For the unrooted spanning-forest measure on any finite simple graph with arbitrary positive edge weights, we prove real stability of the full incident-edge marginal at every vertex. Thus each star is strongly Rayleigh, incident edges are nonpositively correlated, and every vertex degree has a Poisson-binomial law. The proof identifies a classical optional-component-mark forest polynomial with a stable determinant differential operator applied to a signed matching polynomial. We also prove strict pairwise negative edge correlation on every complete graph K_N, N≥3, and complete bipartite graph K_(m,n), m,n≥2, at every common positive fugacity. Their disjoint-edge proofs use exact Hermite and Laguerre counts, continued fractions and scalar crossing barriers, with complete exact polynomial certificates. Quantitative bipartite critical and supercritical deficit asymptotics are also established. Empty forests and isolated vertices are included.
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Research details
- Subjects
- Mathematics · Combinatorics and Graph Theory · Probability and Statistics
- Keywords
- random forests, real stable polynomials, negative dependence, Poisson-binomial laws, arboreal gas
- Manuscript date
- 2026-10-23
- Reader reuse
- No additional permission granted
Scope
Stable incident-edge marginals and Poisson-binomial vertex degrees for arbitrary positively weighted finite simple-graph forest measures, plus strict pairwise negative correlations on complete and complete bipartite hosts at common fugacity.
Disclosures
AI-assisted tools supported research exploration, literature discovery, manuscript preparation and exact verification code. The author is responsible for the mathematical arguments and supporting computations.
Reproduction
29-page manuscript and 23 source-package members. Install requirements.txt with its pinned SymPy dependency, then run python verify.py or python -O verify.py for exact symbolic certificates and finite model checks. Compile the included TeX assembly using build.py and Tectonic. The manuscript separates the arbitrary-weight star law, the two graph-family all-common-fugacity theorem, supporting certificates and quantitative scaling results.