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

Abstract: 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 $\epsilon&gt;0$ 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 \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-\epsilon_0 ) \Delta$, where $\epsilon_0 \approx 10<sup>{-5}$ 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 simplical complex is a local spectral expander.