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

Noela Müller, Ralph Neininger, Haodong Zhu

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.

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

Abstract

We show that the set of atoms of the limiting empirical marginal distribution in the random -SAT model is , for all clause-to-variable densities up to the satisfiability threshold. While for densities up to , the measure is purely discrete, we additionally establish the existence of a nontrivial continuous part for any density in . 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.

Paper Structure

This paper contains 19 sections, 17 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. Main result
  4. Proof strategy
  5. Preliminaries
  6. Recursive computation of tree marginals
  7. Distributional identities from 2sat
  8. Almost sure convergence of the root marginal in a truncated Galton-Watson tree
  9. Multi-type Galton-Watson process
  10. Proof of Proposition \ref{['Prop_asconvergence']}
  11. Convergence of the log-likelihood ratios
  12. Proof of Proposition \ref{['Prop_asconvergence']}
  13. Proof of Proposition \ref{['lem_atoms1']}
  14. Finite rooted trees
  15. Proof of Proposition \ref{['lem_atoms1']}
  16. ...and 4 more sections

Key Result

Proposition 1.1

For any $0<d<2$ there exists a probability distribution $\pi_d$ on $[0,1]$ such that the random probability measure converges to $\pi_d$ weakly in probabilitySpecifically, for any continuous function $f:[0,1]\to\mathbb{R}$, $\lim_{n\to\infty}\mathbb{E}\left|{\int_0^1f(z){\mathrm d}\pi_d(z)-\int_0^1f(z){\mathrm d}\pi_{\boldsymbol{\Phi}}(z)}\right|=0$..

Figures (1)

  • Figure 1: Illustration of the tree $\boldsymbol{T}$ obtained from joining the roots $o_1, o_2,o_3$ of the $\boldsymbol{d}_o=3$ trees $\boldsymbol{T}_1,\boldsymbol{T}_2,\boldsymbol{T}_3$ to a new root variable $o$ via the clauses $\boldsymbol{a}_{o_1}, \boldsymbol{a}_{o_2}, \boldsymbol{a}_{o_3}$ (truncated at level $4$).

Theorems & Definitions (29)

  • Proposition 1.1: 2sat
  • Remark 1.2: 2sat
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Lemma 2.1: 2sat
  • Proposition 3.1
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 19 more