Random $2$-SAT: The set of atoms of the limiting empirical marginal distribution

Noela Müller; Ralph Neininger; Haodong Zhu

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arXiv · 2410.17749

Random $2$-SAT: The set of atoms of the limiting empirical marginal distribution

Noela Müller·Ralph Neininger·Haodong Zhu

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Abstract

We show that the set of atoms of the limiting empirical marginal distribution in the random $2$-SAT model is $\mathbb Q \cap (0,1)$, for all clause-to-variable densities up to the satisfiability threshold. While for densities up to $1/2$, the measure is purely discrete, we additionally establish the existence of a nontrivial continuous part for any density in $(1/2, 1)$. Our proof is based on the construction of a random variable with the correct distribution as the the root marginal of a multi-type Galton-Watson tree, along with a subsequent analysis of the resulting almost sure recursion.

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Noela Müller, Ralph Neininger, Haodong Zhu. 2024-10-23. Random $2$-SAT: The set of atoms of the limiting empirical marginal distribution. https://arxiv.org/abs/2410.17749

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