The Markov gap in the presence of islands

The Markov gap \cite{Hayden:2021gno}, namely the difference between reflected entropy and mutual information, is explicitly computed in the defect extremal surface model, JT gravity, and the generic 2d extremal black holes, in vacuum states. The phases that contain various island contributions are considered, and their existence is carefully checked. Moreover, we show explicitly how the Markov gap originates from the OPE coefficient of the boundary CFT. And, as a generalization of \cite{Hayden:2021gno}, the lower bound of the Markov gap is given by $\frac{c}{3}\log 2$ times the number of EWCS boundaries on minimal surfaces. We propose a boundary way of counting the lower bound for the Markov gap, which states that the lower bound is given by $\frac{c}{3}\log 2$ times the number of gaps between two boundary regions in vacuum states. We discuss the limitation and possible generalization of the boundary counting, and its relation to tripartite entanglement.