We study the relationship between the underlying structure of posets and the spectral and combinatorial properties of their higher-order random walks. While fast mixing of random walks on hypergraphs has led to myriad breakthroughs throughout theoretical computer science in the last five years, many other important applications (e.g. locally testable codes, 2-2 games) rely on the more general non-simplicial structures. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different architectures, highlighting how structural regularity controls the spectral decay and edge-expansion of corresponding random walks. In particular, we show the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the poset's regularity. This gives a simple condition to identify architectures (e.g. the Grassmann) that exhibit fast (exponential) decay of eigenvalues, versus architectures like hypergraphs with slow (linear) decay -- a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight variance-based characterization of edge-expansion on eposets generalizing (Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization used in the proof of the 2-2 Games Conjecture which relies on $\ell_\infty$ rather than $\ell_2$-structure.

翻译：我们研究偏序集底层结构与其高阶随机游走的谱特性和组合特性之间的关系。尽管超图上随机游走的快速混合在过去五年间推动了理论计算机科学领域的诸多突破，但许多其他重要应用（如局部可测试编码、2-2游戏）依赖于更一般的非单纯结构。这些工作表明，偏序集的全局扩张性质强烈依赖于其底层架构（如单纯复形、立方复形、线性代数结构），但整体现象仍未被充分理解。在本研究中，我们量化不同架构的优势，揭示结构规律性如何控制相应随机游走的谱衰减和边扩张。特别地，我们证明扩张偏序集（Dikstein、Dinur、Filmus、Harsha，RANDOM 2018）上随机游走的谱集中在由偏序集规律性控制的少量近似特征值周围的条带中。这给出了一个简单条件来识别具有快速（指数级）特征值衰减的架构（如格拉斯曼流形），与超图等具有缓慢（线性）衰减的架构形成对比——这一区别在近似硬度与一致性测试（如近期2-2游戏猜想的证明，Khot、Minzer、Safra，FOCS 2018）等应用中至关重要。我们证明这些结果导出了对e-偏序集上边扩张的紧致方差刻画，推广了Bafna、Hopkins、Kaufman与Lovett（SODA 2022）的工作，并特别关注格拉斯曼流形情形——我们证明了对于格拉斯曼图的一组自然稀疏化，所得结果是紧的。需澄清的是，我们的结果并未恢复2-2游戏猜想证明中使用的刻画，因其依赖于$\ell_\infty$结构而非$\ell_2$结构。