Sharp √n asymptotic for dense random circulant graphs

Prove that the expected Lovász number of a dense random circulant graph satisfies E[θ(G)]=(1+o(1))√n as the number of vertices tends to infinity.

Background

The main theorem establishes the lower bound √n and an upper bound of order √(n log log n) for the expected Lovász number of a dense random circulant graph. Numerical observations motivate the conjecture that the lower bound gives the exact first-order asymptotic. Establishing this conjecture would remove the remaining √(log log n) factor.