The number of solutions for random regular NAE-SAT

Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of...

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Bibliographic Details
Published in:arXiv.org 2023-11
Main Authors: Sly, Allan, Sun, Nike, Zhang, Yumeng
Format: Article
Language:English
Subjects:
Broken symmetry
Free energy
Mathematical models
Online Access:Get full text
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Summary:Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of these models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain "survey propagation model" associated to this problem. We believe that these methods may be applicable in a wide class of related problems.
ISSN:2331-8422