## SciPost Submission Page

# The asymptotics of the clustering transition for random constraint satisfaction problems

### by Louise Budzynski, Guilhem Semerjian

### Submission summary

As Contributors: | Louise Budzynski |

Arxiv Link: | https://arxiv.org/abs/1911.09377v1 |

Date submitted: | 2019-11-25 |

Submitted by: | Budzynski, Louise |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Statistical and Soft Matter Physics |

Approaches: | Theoretical, Computational |

### Abstract

Random Constraint Satisfaction Problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of $k$-uniform hypergraphs with a density $\alpha$ of constraints, and the $q$-coloring of random graphs with average degree $c$. We show that in the large $k,q$ limit the clustering transition occurs for $\alpha = \frac{2^{k-1}}{k} (\ln k + \ln \ln k + \gamma_{\rm d} + o(1))$, $c= q (\ln q + \ln \ln q + \gamma_{\rm d}+ o(1))$, where $\gamma_{\rm d}$ is the same constant for both models. We characterize $\gamma_{\rm d}$ via a functional equation, solve the latter numerically to estimate $\gamma_{\rm d} \approx 0.871$, and obtain an analytic lowerbound $\gamma_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$. Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at $\gamma_{\rm r}=1$.