  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.
