We study the monomer--dimer partition function on the configuration model of random $d$-regular, $l$-uniform hypergraphs. For fixed $d,l\ge2$, we prove quenched free-energy limits in explicit parameter regimes. The proof combines fixed-density first-moment asymptotics, a two-overlap second-moment variational analysis, and a subgraph-conditioning argument for the short cycles of the incidence structure. The main technical point is to identify regimes in which the replica-symmetric saddle is the unique global maximizer of the second-moment rate function. In those regimes the normalized logarithm of the total matching partition function converges in probability to an explicit variational value. We also prove the corresponding result for the weighted partition function whenever the maximizing density lies in the verified replica-symmetric region, give an additional checkable criterion for that region, and record a first-moment upper tail estimate for the maximum matching size.

翻译：我们研究随机 $d$-正则、$l$-均匀超图配置模型上的单体-二聚体配分函数。对于固定的 $d,l\ge2$，我们在显式参数区域中证明了淬火自由能的极限。证明结合了固定密度的一阶矩渐近分析、双重叠二阶矩变分分析以及针对关联结构中短环的子图条件化论证。主要技术难点在于确定副本对称鞍点是二阶矩率函数唯一全局最大化子的参数区间。在这些区间内，总匹配配分函数的归一化对数依概率收敛于显式变分值。我们还证明了当最大密度位于已验证的副本对称区域内时相应的加权配分函数结果，给出了该区域的一个可检验的附加判据，并记录了最大匹配规模的一阶矩上尾估计。