通过C-洛伦兹多项式为路径复形最优滴落定理
Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences
AI总结：
通过C-洛伦兹多项式，研究路径复形的最优滴落定理，推导出顶链接谱扩张器的快速混合结果和对数凹性，并重新证明Heron-Rota-Welsh猜想及Stanley定理的推广。
摘要：
设X是一个d部分d维的简单复形，其部分为T₁,…,T_d，并设μ是X面的分布。非正式地说，我们称(X,μ)为路径复形，如果对于任何i<j<k以及F∈T_i,G∈T_j,K∈T_k，有P_μ[F,K|G]=P_μ[F|G]·P_μ[K|G]。我们开发了一种新的工具，利用C-洛伦兹多项式来证明，如果X的所有共维数为2的链接的谱扩张至多为1/2，则X是一个1/2-局部谱扩张器。然后我们证明，可以推导出对于顶链接谱扩张器的快速混合结果和对数凹性陈述。我们使用该工具证明了在外部场下，对分布式格子的最大旗 flats 进行快速混合的结果，以及对“典型”模格子的最大旗 flats 进行采样的结果。我们还利用它重新证明了Heron-Rota-Welsh猜想，并证明了Chan和Pak的猜想，该猜想是对Stanley对数凹性定理的推广。最后，我们使用它证明了对于“稀疏复形”如Lubotzky-Samuels-Vishne、Kaufman-Oppenheim和O'Donnell-Pratt的构造的近最优滴落定理。
英文摘要：
Let $X$ be a $d$-partite $d$-dimensional simplicial complex with parts $T_1,\dots,T_d$ and let $μ$ be a distribution on the facets of $X$. Informally, we say $(X,μ)$ 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 $\mathbb{P}_μ[F,K | G]=\mathbb{P}_μ[F|G]\cdot\mathbb{P}_μ[K|G].$ 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.
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