Let $X$ be a $d$-partite $d$-dimensional simplicial complex with parts $T_1,\dots,T_d$ and let $\mu$ be a distribution on the facets of $X$. Informally, we say $(X,\mu)$ is a path complex if for any $i<j<k$ and $F \in T_i,G \in T_j, K\in T_k$, we have $\mathbb{P}_\mu[F,K | G]=\mathbb{P}_\mu[F|G]\cdot\mathbb{P}_\mu[K|G].$ We develop a new machinery with $\mathcal{C}$-Lorentzian polynomials to show that if all links of $X$ of co-dimension 2 have spectral expansion at most $1/2$, then $X$ is a $1/2$-local spectral expander. We then prove that one can derive fast-mixing results and log-concavity statements for top-link spectral expanders. We use our machinery to prove fast mixing results for sampling maximal flags of flats of distributive lattices (a.k.a. linear extensions of posets) subject to external fields, and to sample maximal flags of flats of "typical" modular lattices. We also use it to re-prove the Heron-Rota-Welsh conjecture and to prove a conjecture of Chan and Pak which gives a generalization of Stanley's log-concavity theorem. Lastly, we use it to prove near optimal trickle-down theorems for "sparse complexes" such as constructions by Lubotzky-Samuels-Vishne, Kaufman-Oppenheim, and O'Donnell-Pratt.

翻译：令 $X$ 为一个 $d$ 部 $d$ 维单纯复形，其部分为 $T_1,\dots,T_d$，并令 $\mu$ 为 $X$ 的维面上的一个分布。非正式地说，若对任意 $i<j<k$ 及 $F \in T_i,G \in T_j, K\in T_k$，均有 $\mathbb{P}_\mu[F,K | G]=\mathbb{P}_\mu[F|G]\cdot\mathbb{P}_\mu[K|G]$，则称 $(X,\mu)$ 为一个路径复形。我们发展了一种基于 $\mathcal{C}$-洛伦兹多项式的新工具，证明若 $X$ 的所有余维为 2 的链环的谱展开至多为 $1/2$，则 $X$ 是一个 $1/2$-局部谱展开子。随后我们证明，对于顶链谱展开子，可以推导出快速混合结果与对数凹性陈述。我们利用该工具证明了在外部场约束下，采样分配格（即偏序集的线性扩展）的极大平坦族标志的快速混合结果，以及采样“典型”模格的极大平坦族标志的快速混合结果。此外，我们用它重新证明了 Heron-Rota-Welsh 猜想，并证明了 Chan 与 Pak 提出的一个猜想，该猜想推广了 Stanley 的对数凹性定理。最后，我们用它证明了针对“稀疏复形”（如 Lubotzky-Samuels-Vishne、Kaufman-Oppenheim 及 O'Donnell-Pratt 构造的复形）的近乎最优的滴流定理。