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

2025-03-02

AI Summary

This work addresses spectral properties and sampling efficiency for high-dimensional combinatorial structures. Methodologically, it introduces a novel analytical framework based on $mathcal{C}$-Lorentzian polynomials—the first application of this theory to characterize local spectral expansion in path complexes—and integrates high-dimensional spectral graph theory with Markov chain mixing time analysis to establish a near-optimal bound for the trickle-down theorem ($pi/2 - 1$). Key contributions include: (i) a rigorous proof that path complexes are local spectral expanders; (ii) fast-mixing sampling algorithms for maximal flags on distributive lattices and canonical modular lattices; (iii) establishment of the universality of log-concave sequences; and (iv) a unified reproof of the Heron–Rota–Welsh and Chan–Pak conjectures, along with strengthened combinatorial inequalities. These results substantially advance both the theoretical depth and applicability of Lorentzian methods in high-dimensional combinatorics.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 $X$. We say $(X,mu)$ is a path complex if for any $i<j<k$ and $F in T_i,G in T_j, Kin T_k$, we have $mathbb{P}_mu[F,K | G]=mathbb{P}_mu[F|G]cdotmathbb{P}_mu[K|G].$ 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 $alpha$-local spectral expander for some $alphaleq pi/2-1$. 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 Ramanujan path complexes such as constructions by Lubotzky-Samuels-Vishen, Kaufman-Oppenheim, and O'Donnell-Pratt.

Problem

• Develops $mathcal{C}$-Lorentzian polynomials for path complexes.
• Proves fast mixing for sampling maximal flags in lattices.
• Re-proves and generalizes log-concavity theorems in combinatorics.

Innovation

• Uses $mathcal{C}$-Lorentzian polynomials for analysis
• Proves fast mixing for sampling maximal flags
• Applies to log-concavity and spectral expansion