Spectral gap of random Cayley graphs of the symmetric group

Establish whether random Cayley graphs Cay(S_N;σ_1, …, σ_r), formed by r independent uniformly random generators in S_N, have a nonvanishing spectral gap with high probability as N→∞, and determine whether any such gap is optimal.

Background

The paper analyzes random regular graphs and Schreier graphs arising from actions of S_N on k-tuples, proving optimal spectral gaps for large families via strong convergence. In contrast, the case k=N corresponds to random Cayley graphs of S_N.

The authors note that even the existence of a nonzero spectral gap for these random Cayley graphs is a long-standing open question, highlighting a major gap between known results and this important regime.