bibkey: miaoyin2026matchings authors: Tianshun Miao; Yitong Yin year: 2026 title: Sampling Matchings in Near-linear Time doi: null url: https://arxiv.org/abs/2609.21936v1 claim: Unit-deletion insertion dynamics on a positive downward-closed law gives exact monomial covariance and energy matrices; low-degree approximation bounds additionally require field-dynamics spectral stability and an integrable rate. strata_touched: [] license: citation-only triage: anchor

Monomial forms and the field-dynamics approximation boundary

The primary source is arXiv:2609.21936v1, Section 3.2, equations (13), (21) and (22). The source’s principal sampling results concern matchings. The monomial calculation in Section 3.2 applies to a general positive law on a downward-closed finite configuration family.

Its auxiliary insertion-deletion dynamics deletes every occupied coordinate at rate one and inserts a legal coordinate with rate equal to the occupied-to-empty mass ratio. Equation (13) counts each edge through its occupied endpoint. At retention one, the normalized monomial formulas in (21)–(22) give, after removing the normalizations,

$$\mathrm{Cov}(m_I,m_J)={\mu }_{I\cup J}-{\mu }_I{\mu }_J,{\mathcal{E}}_1(m_I,m_J)=|I\cap J|{\mu }_{I\cup J}.$$

For the product-activity weighted heat-bath form, the deletion rate at site $i$ is instead ${\omega }_i/(1+{\lambda }_i)$. The same occupied endpoint calculation replaces the intersection cardinality by ${\sum }_{i\in I\cap J}{\omega }_i/(1+{\lambda }_i)$. This is a weighted application of the published mechanism, not a new general monomial energy principle. No independence of occupations is assumed.

Lemma 3.2’s quantitative approximation estimate uses Definition 1.1’s covariance domination for every positive pinning of every decreasing-field posterior. The rate must be nonincreasing, at least one, and have finite integral of $(C(t)-1)/(1-t)$. A legal insertion-ratio bound also enters the comparison with uniform-coordinate heat-bath dynamics. These constants have not been bounded uniformly for arbitrary positive FIB activities and site weights. The auxiliary unit-deletion clock must not be substituted for the FIB normalized heat-bath clock.

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