Sharp Lovasz-Theta Bounds on Random Graphs

Abstract

It is well known that the \Lovasz-Theta function of a random graph $G(n,\tfrac{1}{2})$ is $\Theta(\sqrt{n})$. More precisely, it is tightly concentrated in the interval \( [\sqrt{n},\, 2\sqrt{n}], \) where the upper bound follows from an explicit dual witness for the associated semidefinite program. Numerical evidence and heuristic arguments suggest that the true value is $(1+o(1))\sqrt{n}$. However, closing this gap has remained a longstanding challenge, resisting existing techniques even in light of recent progress on sharp algorithmic thresholds and non-asymptotic free probability. In this work, we resolve this question by proving that the \Lovasz-Theta function of $G(n,\tfrac{1}{2})$ is $(1+o_n(1))\sqrt{n}$ with high probability, determining its asymptotic value up to vanishing relative error.