Universal local-limit conjecture for mandatory and blocking edges

Under the setting and notation of Theorem~, let $\mathcal{M}_{\max}(G_n)$ be the set of maximum-size matchings on $G_n$. For each directed edge $\overset{\rightarrow}{e}$, let $i(\overset{\rightarrow}{e})$ denote the associated directed-edge variable, let $\overset{\leftarrow}{e}$ denote the reverse orientation, and let $k$ be the threshold parameter from that setting.

Universal mandatory and blocking edge conjecture. The marked random graphs

$$\left(G_n,\mathbbm{1}_{\bigcap_{M\in\mathcal{M}_{\max}(G_n)}M},\mathbbm{1}_{\bigcap_{M\in\mathcal{M}_{\max}(G_n)}M^{\complement}}\right)$$

converge in the Benjamini–Schramm sense to

$$\left(\mathbb{T},o,\mathbbm{1}_{i(\overset{\rightarrow}{e})+i(\overset{\leftarrow}{e})<k},\mathbbm{1}_{i(\overset{\rightarrow}{e})+i(\overset{\leftarrow}{e})>k}\right).$$

The conjecture asserts that the local geometry of mandatory and blocking edges is universal in the sense suggested by the preceding result: its limiting law depends on the unweighted tree rather than on the weight distribution. The precise construction of the variables and the hypotheses are contained in the referenced theorem and propositions.