The Ising $p$-spin glass and the random $k$-SAT models exhibit symmetric multi Overlap Gap Property ($m$-OGP), an intricate geometrical property which is a rigorous barrier against many important classes of algorithms. We establish that for both models, the symmetric $m$-OGP undergoes a sharp phase transition and we identify the phase transition point for each model: for any $m\in\mathbb{N}$, there exists $\gamma_m$ (depending on the model) such that the model exhibits $m$-OGP for $\gamma>\gamma_m$ and the $m$-OGP is provably absent for $\gamma<\gamma_m$, both with high probability, where $\gamma$ is some natural parameter of the model. Our results for the Ising $p$-spin glass model are valid for all large enough $p$ that remain constant as the number $n$ of spins grows, $p=O(1)$; our results for the random $k$-SAT are valid for all $k$ that grow mildly in the number $n$ of Boolean variables, $k=\Omega(\ln n)$. To the best of our knowledge, these are the first sharp phase transition results regarding the $m$-OGP. Our proofs are based on an application of the second moment method combined with a concentration property regarding a suitable random variable. While a standard application of the second moment method fails, we circumvent this issue by leveraging an elegant argument of Frieze~\cite{frieze1990independence} together with concentration.

翻译：Ising $p$-自旋玻璃和随机$k$-SAT模型展现出对称的多重交叠间隙性质（$m$-OGP），这是一类复杂的几何特性，对许多重要算法类别构成了严格的障碍。我们证明，对于这两个模型，对称$m$-OGP经历了一个尖锐的相变，并确定了每个模型的相变点：对任意$m\in\mathbb{N}$，存在$\gamma_m$（依赖于模型），使得当$\gamma>\gamma_m$时模型以高概率表现出$m$-OGP，而当$\gamma<\gamma_m$时$m$-OGP以高概率被证伪，其中$\gamma$是模型的某个自然参数。我们对Ising $p$-自旋玻璃模型的结果适用于所有足够大的$p$，且$p$在自旋数$n$增长时保持恒定，即$p=O(1)$；对随机$k$-SAT模型的结果适用于所有$k$，且$k$在布尔变量数$n$中温和增长，即$k=\Omega(\ln n)$。据我们所知，这是关于$m$-OGP的首批尖锐相变结果。我们的证明基于二阶矩方法的应用，并结合了关于某个合适随机变量的集中性质。尽管二阶矩方法的标准应用失败，我们通过利用Frieze~\cite{frieze1990independence}的一个优雅论点及集中性来规避这一问题。