Sharp Phase Transition for Multi Overlap Gap Property in Ising $p$-Spin Glass and Random $k$-SAT Models

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.