Research Article · Preprint Mathematics
Degree laws and edge correlations for random forests
- Primary subject
- Mathematics
- Subject subfields
- Combinatorics and Graph Theory · Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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 retained. Empty forests and isolated vertices are included.
Keywords Combinatorics and Graph Theory · Probability and Statistics · mathematical research
Scope of the work
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.
Reproduction notes
28-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 to compare the exact forest-polynomial certificate files. --write-certificates regenerates the five generated files from exact formulas. Compile the included TeX assembly using build.py and Tectonic.
The manuscript
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The complete manuscript, as published in this edition.
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Author disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Degree laws and edge correlations for random forests (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/degree-laws-and-edge-correlations-for-random-forests/v/2
BibTeX
@techreport{luminary_degree_laws_and_edge_correlations_for_random_forests_v2,
title = {Degree laws and edge correlations for random forests},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/degree-laws-and-edge-correlations-for-random-forests/v/2}
}Loading discussion…