Optimal Trickle-Down Theorems for Path Complexes via C-Lorentzian Polynomials with Applications to Sampling and Log-Concave Sequences

TL;DR

The paper develops a unified framework connecting C-Lorentzian polynomials with path and top-link path complexes to obtain optimal trickle-down spectral bounds. By constructing commutative π-maps, it proves that 1/2-top-link expanders imply 1/2-local spectral expansion, and derives fast-mixing results for sampling maximal flags in distributive/modular lattices and matroids, along with log-concavity results such as the Heron-Rota-Welsh conjecture and generalized Stanley-type inequalities. It further extends these techniques to sparse, top-link path complexes, providing near-optimal spectral bounds and lower-bound demonstrations to establish optimality. The work bridges Hodge-theoretic and polynomial-geometry methods with Markov-chain analysis to yield new sampling/counting tools and deep log-concavity insights across lattices and matroids. The resulting framework offers both theoretical and algorithmic advances with broad applicability to combinatorial sampling and counting problems.

Abstract

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.

Figures (5)

• Figure 1: All possible distributive lattices for which $\bm{\mathrm{rk}}(\hat{1})=3.$
• Figure 2: $y^*$ has degree 2; neighbors of $x_1,x_2$ partition the vertices of $Y-\{y^*\}$
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Theorems & Definitions (139)

• Definition 1.1: (Top-link) Spectral Expanders
• Theorem 1.2: Oppenheim, Opp18
• Theorem 1.3: AO23
• Theorem 1.4: Trickle-down Theorem for Path Complexes
• Theorem 1.4: Trickle-down Theorem for Path Complexes
• Theorem 1.5: Lower Bound
• Theorem 1.6: Main Algorithmic
• Definition 1.8: Distributive Lattices
• Theorem 1.9
• Definition 1.10: Modular Lattices