Sharp Phase Transition for Multi Overlap Gap Property in Ising p-Spin Glass and Random k-SAT Models
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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∈ℕ, there exists γ_m (depending on the model) such that the model exhibits m-OGP for γ>γ_m and the m-OGP is provably absent for γ<γ_m, both with high probability, where γ 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=Ω(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 <cit.> together with concentration.
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