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Local geometry of NAE-SAT solutions in the condensation regime

Allan Sly, Youngtak Sohn

TL;DR

This work analyzes the local geometry of typical NAE-SAT solutions in the condensation regime for random regular graphs. It proves a non-Markovian local weak limit in the 1RSB class by coupling finite-radius neighborhoods to a broadcast process determined by the 1RSB BP fixed point, and provides a non-asymptotic $O(n^{-1/2})$ fluctuation description. The approach hinges on a detailed analysis of free components and their profiles, exponential decay of large free components, and a refined $ ext{l}^1$-type concentration, culminating in a robust coupling from 1-neighborhoods to t-neighborhoods for fixed t. The results establish the first rigorous local weak limit description in the condensation regime for sparse random CSPs in the 1RSB class and suggest that the technique extends to a broad class of models within this universality class.

Abstract

The local behavior of typical solutions of random constraint satisfaction problems (CSP) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. The work of Coja-Oghlan, Kapetanopoulos, Müller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular NAE-SAT model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters. This is the first description of the local weak limit in the condensation regime for any sparse random CSPs in the one-step replica symmetry breaking (1RSB) class. Our result is non-asymptotic, and characterizes the tight fluctuation $O(n^{-1/2})$ around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1RSB belief-propagation fixed point. We believe that our proof technique has broad applicability to random CSPs in the 1RSB class.

Local geometry of NAE-SAT solutions in the condensation regime

TL;DR

This work analyzes the local geometry of typical NAE-SAT solutions in the condensation regime for random regular graphs. It proves a non-Markovian local weak limit in the 1RSB class by coupling finite-radius neighborhoods to a broadcast process determined by the 1RSB BP fixed point, and provides a non-asymptotic fluctuation description. The approach hinges on a detailed analysis of free components and their profiles, exponential decay of large free components, and a refined -type concentration, culminating in a robust coupling from 1-neighborhoods to t-neighborhoods for fixed t. The results establish the first rigorous local weak limit description in the condensation regime for sparse random CSPs in the 1RSB class and suggest that the technique extends to a broad class of models within this universality class.

Abstract

The local behavior of typical solutions of random constraint satisfaction problems (CSP) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. The work of Coja-Oghlan, Kapetanopoulos, Müller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular NAE-SAT model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters. This is the first description of the local weak limit in the condensation regime for any sparse random CSPs in the one-step replica symmetry breaking (1RSB) class. Our result is non-asymptotic, and characterizes the tight fluctuation around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1RSB belief-propagation fixed point. We believe that our proof technique has broad applicability to random CSPs in the 1RSB class.
Paper Structure (23 sections, 27 theorems, 162 equations, 3 figures)

This paper contains 23 sections, 27 theorems, 162 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and Main result
  3. Related works
  4. Clusters
  5. Local weak limit
  6. Further discussion
  7. Proof overview
  8. Free components
  9. Exponential decay of the frequencies of free components
  10. Tight concentration of free component profile
  11. From $1$-neighborhoods to $t$-neighborhoods
  12. Concentration of coloring profile in $\ell^1$-type distance
  13. Exponential decay of free component profile
  14. Concentration of free tree profile
  15. Coupling $t$-neighborhoods to a broadcast process on a tree
  16. ...and 8 more sections

Key Result

Theorem 1.1

(Informal) For $k\geq k_0$ and $\alpha \in (\alpha_{\textsf{cond}}(k),\alpha_{\textsf{sat}}(k))$, consider a random regular $k\textsc{-nae-sat}$ solution $\underline{\textbf{z}}\in \{0,1\}^n$ defined in Section subsec:def. For $t\geq 1$, the empirical distribution over balls of radius $2t$ of the so

Figures (3)

  • Figure 1: Figure adapted fromkmrsz07dss22. A pictorial description of the conjectured phase diagram of random constraint satisfaction problems in the one-step replica symmetry breaking class. In the condensation regime $(\alpha_{\textsf{cond}},\alpha_{\textsf{sat}})$, a bounded number of clusters contain most of the solutions and the uniform measure over the solutions fails to be contiguous with the planted model.
  • Figure 2: $\mathscr{T}_{d,k,t}$ for $d=k=3$ and $t=2$. Variables and clauses are drawn by the circular and square nodes, respectively. The boundary half-edges in $\partial \mathscr{T}_{d,k,t}$ are highlighted in blue.
  • Figure 3: A pictorial description of a scenario where two distinct leaves $v_1,v_2$ of $N_t(i,\mathcal{G})$ are connected by a path (colored yellow), which lies inside $\mathfrak{f}$ and outside $N_t(i,\mathcal{G})$. The blue curves represent the free component $\mathfrak{f}$ intersecting with $N_t(i,\mathcal{G})$. The red dashed line represents the edges colored $\{{{\scriptsize{\texttt{R}}}}_0,{{\scriptsize{\texttt{R}}}}_1,{{\scriptsize{\texttt{B}}}}_0,{{\scriptsize{\texttt{B}}}}_1,{\scriptsize{\texttt{S}}}\}$ under the component coloring $\underline{\sigma}$. Lemma \ref{['lem:bdry:cycle']} shows that most of the cycles contain more than $2t$ edges colored $\{{{\scriptsize{\texttt{R}}}}_0,{{\scriptsize{\texttt{R}}}}_1,{{\scriptsize{\texttt{B}}}}_0,,{{\scriptsize{\texttt{B}}}}_1,{\scriptsize{\texttt{S}}}\}$.

Theorems & Definitions (72)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5: Coarsening and Frozen configuration
  • Proposition 1.6: Proposition 1.2 in ssz22
  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 62 more