## Abstract

In a broad class of sparse random constraint satisfaction problems (CSP), deep heuristics from statistical physics predict that there is a condensation phase transition before the satisfiability threshold, governed by one-step replica symmetry breaking (1RSB). In fact, in random regular k-NAE-SAT, which is one of such random CSPS, it was verified [1] that its free energy is well-defined and the explicit value follows the 1RSB prediction. However, for any model of sparse random CSP, it has been unknown whether the solution space indeed condensates on O(1) clusters according to the 1RSB prediction. In this paper, we give an affirmative answer to this question for the random regular k-NAE-SAT model. Namely, we prove that with probability close to one, most of the solutions lie inside a bounded number of solution clusters whose sizes are comparable to the scale of the free energy. Furthermore, we establish that the overlap between two independently drawn solutions concentrates precisely at two values. This is the defining property of the one-step replica symmetry breaking class which we establish for the first time in a sparse random CSP, Our proof is based on a detailed moment analysis of a spin system, which has an infinite spin space that encodes the structure of solution clusters. We develop new techniques to study the partition function as well as enhance previous approaches which were only applicable to spin systems with finitely many spins. We believe that our method is applicable to a broad range of random CSPS in the 1RSB universality class.

Original language | English (US) |
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Title of host publication | Proceedings - 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science, FOCS 2021 |

Publisher | IEEE Computer Society |

Pages | 310-318 |

Number of pages | 9 |

ISBN (Electronic) | 9781665420556 |

DOIs | |

State | Published - 2022 |

Event | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 - Virtual, Online, United States Duration: Feb 7 2022 → Feb 10 2022 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2022-February |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021 |
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Country/Territory | United States |

City | Virtual, Online |

Period | 2/7/22 → 2/10/22 |

## All Science Journal Classification (ASJC) codes

- Computer Science(all)

## Keywords

- condensation phase transition
- one-step replica symmetry breaking
- random constraint satisfaction problems
- random regular NAE-SAT model