发表机构
King’s College London; Georgia Institute of Technology; Max Planck Institute for Software Systems; Hebrew University of Jerusalem（伦敦国王学院; 佐治亚理工学院; 马克斯·普朗克软件系统研究所; 希伯来大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了一个变分公式，用于描述随机图中避免团等组合问题中非存在概率的对数渐近，该公式在整个临界区域成立，并基于超图硬核模型的副本对称Bethe自由能，通过窗口熵方法和信念传播引导的贪婪算法证明。
AI 中文摘要
我们证明了一个变分公式，用于描述一类广泛组合问题中非存在概率的对数渐近行为，例如在随机图中避免团以及在整数的随机子集中避免$k$项等差数列。这些结果源于一个公式，该公式给出局部稀疏超图的顶点二项随机子集是独立集的概率。该公式在整个临界区域成立，在Janson不等式和超图容器方法分别给出相应渐近行为的区域之间进行插值。该公式是统计物理学中应用于超图硬核模型的副本对称Bethe自由能公式。证明使用了两种新技术。第一种用于上界，涉及揭示随机独立集的一个小“窗口”，然后应用熵方法。第二种用于下界，涉及分析由信念传播不动点引导的随机贪婪算法。在随机图中避免团的情形下，变分公式可以表示为关于图同态的优化问题。此外，在适当归一化后，以不包含$K_r$为条件的典型随机图，在切割距离下趋近于优化图同态的集合。
英文摘要
We prove a variational formula for the logarithmic asymptotics of a non-existence probability in a broad class of combinatorial problems such as avoiding cliques in random graphs and $k$-term arithmetic progressions in random subsets of integers. These results follow from a formula for the probability that a binomial random subset of the vertices of a locally sparse hypergraph is an independent set. The formula holds throughout the critical regime, interpolating between the regimes in which Janson's inequality and the method of hypergraph containers give the respective asymptotics. The formula is the replica-symmetric Bethe free energy formula from statistical physics applied to a hypergraph hardcore model. The proof uses two new techniques. The first, for the upper bound, involves revealing a small `window' of a random independent set and then applying entropy methods. The second, for the lower bound, involves the analysis of a random greedy algorithm guided by a Belief Propagation fixed point. In the case of avoiding cliques in random graphs, the variational formula can be expressed as an optimization problem over graphons. Moreover, typical random graphs conditioned on not containing $K_r$, when suitably normalized, approach the set of optimizing graphons in cut distance.
Comments48 pages
