Based on joint work with Davide Macera and Joe Thomas. The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about the surface. We study the size of the spectral gap for random large genus hyperbolic surfaces sampled according to the Weil-Petersson probability measure. We show that there is a c>0 such that a random surface of genus g has spectral gap at least 1/4-O(g^-c) with high probability.  Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel, and its generalization to the strong convergence of premutation representations of surface groups by Magee, Puder and van Handel, to the Laplacian on Weil-Petersson random hyperbolic surfaces.