Expansion for random Cayley graphs of the symmetric group

Establish that, for a fixed number r of uniformly random generators, the Cayley graph Cay(S_N; σ_1,…,σ_r) is an expander with high probability as N→∞; equivalently, show that the maximum nontrivial eigenvalue is bounded away from the trivial eigenvalue in this setting, and ultimately determine whether these graphs achieve the optimal spectral gap 2√(2r−1).

Background

The paper studies random Schreier graphs built from actions of S_N and highlights the extreme case k=N, which corresponds to random Cayley graphs of S_N. While optimal spectral gaps are known for random regular graphs, much less is known for random Cayley graphs of S_N generated by a fixed number of random elements.

The authors point out that even the basic expansion property is not proven in this regime, underscoring a significant gap in current techniques.