Exact asymptotic behavior of the Lovász number for dense random circulant graphs

Characterize the exact asymptotic behavior of the Lovász number of a dense random circulant graph, beyond the established bounds [1m[0m√n[0m≤E[θ(G)]≤C√(n log log n).

Background

Dense random circulant graphs are highly structured Cayley graphs whose adjacency matrices are diagonalized by the discrete Fourier transform. The paper proves that their expected Lovász number is between √n and C√(n log log n), leaving a logarithmic-logarithmic gap. The abstract explicitly identifies the exact behavior as unresolved, while the discussion formulates a sharper conjecture addressing this gap.