发表机构
University of Chicago; Toyota Technological Institute at Chicago（芝加哥大学; 芝加哥丰田技术研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明随机图$G(n,1/2)$的Lovász-Theta函数以高概率为$(1+o(1))\sqrt{n}$，解决了长期存在的渐近值精确确定问题。
AI 中文摘要
众所周知，随机图$G(n,\tfrac{1}{2})$的Lovász-Theta函数为$\Theta(\sqrt{n})$。更精确地说，它紧密集中在区间\\( [\sqrt{n},\\, 2\sqrt{n}] \\)内，其中上界来自相关半定规划的显式对偶见证。数值证据和启发式论证表明真实值为$(1+o(1))\sqrt{n}$。然而，弥合这一差距一直是一个长期挑战，即使近期在尖锐算法阈值和非渐近自由概率方面取得了进展，现有技术仍无法解决。在本工作中，我们通过证明$G(n,\tfrac{1}{2})$的Lovász-Theta函数以高概率为$(1+o_n(1))\sqrt{n}$来解决此问题，从而在相对误差趋于零的意义下确定了其渐近值。
英文摘要
It is well known that the \Lovasz-Theta function of a random graph $G(n,\tfrac{1}{2})$ is $Θ(\sqrt{n})$. 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.
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