Spectral gaps on thick part of moduli spaces

In this paper, we study spectral gaps of closed hyperbolic surfaces for large genus. We show that for any fixed $k\ge 1$, as the genus goes to infinity, the maximum of ${\lambda }_k-{\lambda }_{k-1}$ over any thick part of the moduli space of closed Riemann surfaces approaches the limit $\frac{1}{4}$.