Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

TL;DR

This paper explores how the structure of posets influences the spectral and combinatorial properties of their higher-order random walks, revealing how architectural regularity affects expansion and eigenvalue decay, with implications for theoretical computer science.

Contribution

It provides a spectral analysis framework for posets, identifying how different architectures control eigenvalue decay and expansion, and offers a variance-based characterization applicable to various poset structures.

Findings

Spectra of walks on expanding posets concentrate around a few eigenvalues.

Architectures like Grassmann exhibit exponential eigenvalue decay, unlike hypergraphs.

Results are tight for Grassmann sparsifications and relate to hardness of approximation applications.

Abstract

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.