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The random $k$-SAT Gibbs uniqueness threshold revisited

Arnab Chatterjee, Amin Coja-Oghlan, Catherine Greenhill, Vincent Pfenninger, Maurice Rolvien, Pavel Zakharov, Kostas Zampetakis

TL;DR

The paper rigorously verifies the replica symmetric prediction for the exponential growth rate of satisfying assignments in random k-SAT up to the Gibbs uniqueness threshold, by tying the limiting log-partition function to a Bethe free entropy computed via a Belief Propagation fixed point. It introduces an enhanced lower bound on the Gibbs uniqueness threshold through a refined contraction argument that explicitly accounts for pure literals, and it develops a robust Aizenman–Sims–Starr scheme combined with a novel Pure Literal Pursuit (PULP) analysis to obtain sharp upper and matching lower bounds. The work integrates interpolation, boundary-condition analysis on Galton–Watson trees, and detailed tail control of hard constraints to bridge physics predictions and rigorous combinatorial counting. The results yield the first rigorous RS formula for a non-trivial regime of densities in random k-SAT and offer substantive improvements on d_uniq(k) for small k, with implications for counting, sampling, and algorithmic performance near the satisfiability threshold.

Abstract

We prove that for any $k\geq3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the `replica symmetric solution' predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)].The improvement is significant particularly for small $k$.

The random $k$-SAT Gibbs uniqueness threshold revisited

TL;DR

The paper rigorously verifies the replica symmetric prediction for the exponential growth rate of satisfying assignments in random k-SAT up to the Gibbs uniqueness threshold, by tying the limiting log-partition function to a Bethe free entropy computed via a Belief Propagation fixed point. It introduces an enhanced lower bound on the Gibbs uniqueness threshold through a refined contraction argument that explicitly accounts for pure literals, and it develops a robust Aizenman–Sims–Starr scheme combined with a novel Pure Literal Pursuit (PULP) analysis to obtain sharp upper and matching lower bounds. The work integrates interpolation, boundary-condition analysis on Galton–Watson trees, and detailed tail control of hard constraints to bridge physics predictions and rigorous combinatorial counting. The results yield the first rigorous RS formula for a non-trivial regime of densities in random k-SAT and offer substantive improvements on d_uniq(k) for small k, with implications for counting, sampling, and algorithmic performance near the satisfiability threshold.

Abstract

We prove that for any for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random -SAT formulas is given by the `replica symmetric solution' predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any , we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)].The improvement is significant particularly for small .

Paper Structure

This paper contains 33 sections, 45 theorems, 221 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Results
  4. The number of satisfying assignments up to the Gibbs uniqueness threshold
  5. An improved lower bound on Gibbs uniqueness
  6. Preliminaries and notation
  7. Overview
  8. Existence of the fixed point and upper bound
  9. A matching lower bound
  10. Pure literal pursuit
  11. Completing the Aizenman-Sims-Starr scheme
  12. Lower-bounding the uniqueness threshold
  13. The extremal boundary condition
  14. Pure and mixed literals
  15. Coupling and contraction
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $k\geq3$ and assume that $0<d<d_{\mathrm{uniq}}(k)$. Then the weak limit exists and

Figures (3)

  • Figure 1: Comparison of $\mathfrak B_{d,k}(\pi_{d,k})$ with known bounds for $\lim_{n\to\infty}\frac{1}{n}\log Z(\boldsymbol{\Phi})$ for $k=3$. The red dotted line depicts the first moment upper bound \ref{['eq1mmt']}, while the green dotted line represents the lower bound provided by \ref{['eq2mmt']}. The blue line displays a numerical approximation of $\mathfrak B_{d,3}(\pi_{d,3})$. To obtain our values, we generated $10^{6}$ samples from $\pi \approx \mathrm{BP}^{25}_{d,3}(\delta_{1/2})$ and then evaluated the corresponding empirical average of the expression in \ref{['eqBethe']}.
  • Figure 2: Example of a coupling between derivative terms in \ref{['eqtypeContr2']}--\ref{['eqtypeContr2N']}. For vector $r$ and type $t\in \{\mathrel{\raisebox{0.3pt}{$\CIRCLE$}}, \mathrel{\raisebox{0pt}{$\oplus$}}, \mathrel{\raisebox{0pt}{$\ominus$}}\}$, we pair the term $\mathcal{D}^{t}(z, r; +1)$ in \ref{['eqtypeContr2']} with the term $\mathcal{D}^{t}(z, \mathfrak{p}_t(r); -1)$ in \ref{['eqtypeContr2N']}.
  • Figure 3: A sketch depicting the subformulas $\boldsymbol{\psi}^+, \boldsymbol{\psi}^-, \boldsymbol{\phi}'_{\Lambda,L}$, and $\boldsymbol{\phi}'_{\Lambda^+,L}$ of $\boldsymbol{\Phi}'$ constructed above.

Theorems & Definitions (97)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 1.4: Bennett's inequality LugCon
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 87 more