Jiaoyang Huang - Harvard University I will discuss the extreme eigenvalue distributions of sparse Erdos–Rényi graphs G(N,p). We prove that there is a crossover in the behavior of the extreme eigenvalues at p∼N−2/3. When p≫N−2/3, the extreme eigenvalues have asymptotically Tracy-Widom fluctuations. When N−7/9≪p≪N−2/3, the extreme eigenvalues have asymptotically Gaussian fluctuations. When p=CN−2/3, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy-Widom distribution. Our proof is based on constructing a higher order self-consistent equation for the Stieltjes transform of the empirical eigenvalue distributions. This is based on joint work with Benjamin Landon and Horng-Tzer Yau.