A Matrix Trickle-Down Theorem on Simplicial Complexes and Applications to Sampling Colorings

We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree
$\Delta$
whenever the number of colors is at least
$q\geq(\frac{10}{3}+\epsilon)\Delta$
, where
$\epsilon > 0$
is arbitrary and the maximum degree satisfies
$\Delta\geq C$
for a constant
$C=C(\epsilon)$
depending only on
$\epsilon$
, For edge-colorings, this improves upon prior work [Vig99; Che+19] which show rapid mixing when
$q\geq(\frac{11}{3}-\epsilon_{0})\Delta$
, where
$\epsilon_{0}\approx 10^{-5}$
is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenheim's influential result, as a new technique to prove that a high dimensional simplicial complex is a local spectral expander.