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, the full incident-edge marginal at every vertex is real stable. Each star is strongly Rayleigh, incident edges are nonpositively correlated, and vertex degrees have Poisson-binomial laws. The proof combines a classical optional-component-mark forest polynomial with a stable determinant differential operator and a signed matching polynomial. We also prove strict pairwise negative edge correlation on every complete graph with at least three vertices and every complete bipartite graph with both parts at least two, at every common positive fugacity. Exact Hermite and Laguerre counts, continued fractions, scalar crossing barriers and polynomial certificates treat disjoint edges in these families. Quantitative bipartite critical and supercritical deficit asymptotics are retained. Arbitrary-host disjoint-edge correlation and negative association of the full forest law remain outside the results.
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Research details
- Subjects
- Mathematics · Combinatorics and Graph Theory · Probability and Statistics
- Keywords
- Combinatorics and Graph Theory, Probability and Statistics, mathematical research
- Manuscript date
- 2026-10-23
- Reader reuse
- No additional permission granted
Scope
For every finite simple graph with arbitrary positive edge weights, the full incident-edge marginal at each vertex is real stable/strongly Rayleigh: star negative association, nonpositive incident covariance and Poisson-binomial vertex degrees. Equivalently the optional single-mark-per-component polynomial is stable. Conditioning on required/forbidden edges gives the reviewed contracted-star/component-boundary corollary. Additionally every complete K_N (N>=3) and complete bipartite K_(m,n) (m,n>=2) has strictly negative covariance for all distinct edge pairs at every COMMONpositive fugacity, with stars independent and K1/K2 vacuous. Quantitative bipartite critical/supercritical deficits retain stated scaling domains.
Disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work.
Reproduction
The exact publication package includes the 28-page manuscript, editable LaTeX sources, reader/build instructions, citation metadata, public checksums and any necessary mathematical or computational supplements. Keep its 23 members together and follow the included README for dependencies, build commands and any finite verification programs. Published finite checks support the stated analytic arguments.