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arXiv · 2106.03845

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

Dorna Abdolazimi·Kuikui Liu·Shayan Oveis Gharan
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Abstract

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 $q\geq (\frac{10}{3} + ε)Δ$, where $ε>0$ is arbitrary and the maximum degree satisfies $Δ\geq C$ for a constant $C = C(ε)$ depending only on $ε$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-ε_0 ) Δ$, where $ε_0 \approx 10^{-5}$ 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 simplical complex is a local spectral expander.

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Dorna Abdolazimi, Kuikui Liu, Shayan Oveis Gharan. 2021-11-16. A Matrix Trickle-Down Theorem on Simplicial Complexes and Applications to Sampling Colorings. https://arxiv.org/abs/2106.03845

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