Fixed-degree expanders are sparse yet highly connected graphs. This quality is captured by their spectral gap -- the difference between the largest and second largest eigenvalues of their adjacency
matrix. A celebrated result of Friedman states that a random d-regular graph on n vertices is a near-optimal expander with high probability. I will discuss a generalization of this result to a regime where the number of vertices grows quasi-exponentially in n. The proof draws on ideas from representation theory and considerations of word maps on the symmetric group.