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

TL;DR

This paper characterizes the atoms of the limiting empirical marginal distribution in random 2-SAT, revealing a phase transition from discrete to continuous behavior as clause density increases.

Contribution

It establishes the structure of the limiting distribution's atoms for all clause-to-variable densities up to the satisfiability threshold, including the existence of a continuous part.

Findings

Atoms form the rationals in (0,1) for all densities

Distribution is purely discrete for densities up to 1/2

A nontrivial continuous part exists for densities in (1/2, 1)

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.