Random Cayley Graphs are Expanders: a Simple Proof of the Alon-Roichman Theorem

We give a simple proof of the Alon–Roichman theorem, which asserts that the Cayley graph obtained by selecting cε log |G| elements, independently and uniformly at random, from a finite group G has expected second eigenvalue no more than ε; here cε is a constant that depends only on ε. In particular, such a graph is an expander with constant probability. Our new proof has three advantages over the original proof: (i.) it is extremely simple, relying only on the decomposition of the group algebra and tail bounds for operator-valued random variables, (ii.) it shows that the log |G| term may be replaced with log D, where D ≤ |G| is the sum of the dimensions of the irreducible representations of G, and (iii.) it establishes the result above with a smaller constant cε.