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We show that the natural Glauber dynamics mixes rapidly and generates arandom proper edge-coloring of a graph with maximum degree $\Delta$ wheneverthe number of colors is at least $q\geq (\frac{10}{3} + \epsilon)\Delta$, where$\epsilon>0$ is arbitrary and the maximum degree satisfies $\Delta \geq C$ fora constant $C = C(\epsilon)$ depending only on $\epsilon$. For edge-colorings,this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixingwhen $q\geq (\frac{11}{3}-\epsilon_0 ) \Delta$, where $\epsilon_0 \approx10^{-5}$ is a small fixed constant. At the heart of our proof, we establish amatrix trickle-down theorem, generalizing Oppenheim's influential result, as anew technique to prove that a high dimensional simplical complex is a localspectral expander.

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