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The number of random 2-SAT solutions is asymptotically log-normal

Arnab Chatterjee, Amin Coja-Oghlan, Noela Müller, Connor Riddlesden, Maurice Rolvien, Pavel Zakharov, Haodong Zhu

TL;DR

It is proved that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem, which implies that the log of the number of satisfying assignments exhibits fluctuations of order $n$ with $n$ the number of variables.

Abstract

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order $\sqrt n$, with $n$ the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.

The number of random 2-SAT solutions is asymptotically log-normal

TL;DR

It is proved that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem, which implies that the log of the number of satisfying assignments exhibits fluctuations of order with the number of variables.

Abstract

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order , with the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.
Paper Structure (38 sections, 40 theorems, 163 equations, 2 figures, 1 algorithm)

This paper contains 38 sections, 40 theorems, 163 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The main result
  4. Proof strategy
  5. The method of moments fails
  6. Belief Propagation
  7. Approaching the variance
  8. Unit Clause Propagation
  9. Variance redux
  10. Local convergence in probability
  11. Correlated Belief Propagation
  12. The central limit theorem
  13. Discussion
  14. Preliminaries and notation
  15. Boolean formulas
  16. ...and 23 more sections

Key Result

Theorem 1.1

For any $0<d<2$, $t\in[0,1]$ there exists a unique probability measure $\rho_{d,t}\in\mathcal{P}(\mathbb{R}^2)$ such that Furthermore,

Figures (2)

  • Figure 1: Left: Numerical approximations to the function $\phi(d)$ from \ref{['eqlaregenum']} (red) and the variance $\eta(d)^2$ from \ref{['eqthm_clt']} (green). The black dashed line is the first moment bound $d\mapsto\log(2)+\frac{d}{2}\log(3/4)$. Right: An illustration of the tree $\boldsymbol{T}^\otimes$ from Section \ref{['sec_lwc']}.
  • Figure 2: The distributions $\mathfrak{t}(\rho_{d,t})$ for $d=1.9$ and $t=0.1 ,0.5,0.9$.

Theorems & Definitions (99)

  • Theorem 1.1
  • Proposition 2.1: 2sat
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 89 more