A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts

Abstract: It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most $2\sqrt{d-1} +o(1)$ with probability tending to one as the size of the graph tends to infinity. We give a new proof of this statement. We also study related questions on random $n$-lifts of graphs and improve a recent result by Friedman and Kohler.