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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 $\Delta$ whenever the number of colors is at least $q\geq (\frac{10}{3} + \epsilon)\Delta$, where $\epsilon>0$ is arbitrary and the maximum degree satisfies $\Delta \geq C$ for a constant $C = C(\epsilon)$ depending only on $\epsilon$.
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