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- W4386876284 abstract "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 $minmathbb{N}$, 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." @default.
- W4386876284 created "2023-09-20" @default.
- W4386876284 creator A5029207194 @default.
- W4386876284 date "2023-09-18" @default.
- W4386876284 modified "2023-09-27" @default.
- W4386876284 title "Sharp Phase Transition for Multi Overlap Gap Property in Ising $p$-Spin Glass and Random $k$-SAT Models" @default.
- W4386876284 doi "https://doi.org/10.48550/arxiv.2309.09913" @default.
- W4386876284 hasPublicationYear "2023" @default.
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