A Matrix Trickle-Down Theorem on Simplicial Complexes and Applications to Sampling Colorings
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We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree Δ whenever the number of colors is at least q≥ (10/3 + ϵ)Δ, where ϵ>0 is arbitrary and the maximum degree satisfies Δ≥ C for a constant C = C(ϵ) depending only on ϵ. For edge-colorings, this improves upon prior work <cit.> which show rapid mixing when q≥ (11/3-ϵ_0 ) Δ, where ϵ_0 ≈ 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 simplical complex is a local spectral expander.
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