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

Dorna Abdolazimi, Kuikui Liu, Shayan Oveis Gharan

Abstract: We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Delta$</tex> whenever the number of colors is at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q\geq(\frac{10}{3}+\epsilon)\Delta$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon &gt; 0$</tex> is arbitrary and the maximum degree satisfies <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Delta\geq C$</tex> for a constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C=C(\epsilon)$</tex> depending only on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon$</tex> , For edge-colorings, this improves upon prior work [Vig99; Che+19] which show rapid mixing when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$q\geq(\frac{11}{3}-\epsilon_{0})\Delta$</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon_{0}\approx 10^{-5}$</tex> 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.