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In this paper we introduce the two-sided level-set for the two-dimensionaldiscrete Gaussian free field. Then we investigate the chemical distance for thetwo-sided level-set percolation. Our result shows that the chemical distanceshould have dimension strictly larger than $1$, which in turn stimulates sometempting questions about the two-sided level-set.

On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field