The asymptotics of the clustering transition for random constraint satisfaction problems

TL;DR

This paper analyzes the asymptotic behavior of the clustering transition in random constraint satisfaction problems, revealing precise thresholds and their relation to other phase transitions in large parameter limits.

Contribution

It derives the asymptotic thresholds for clustering in hypergraph bicoloring and graph q-coloring, and characterizes the critical constant via a functional equation.

Findings

Clustering transition occurs at specific thresholds in large k,q limits.

The critical constant b3_d is approximately 0.871, with a proven lower bound of about 0.812.

The clustering transition is closely linked to the rigidity threshold at b3_r=1.

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 $\gamma_{\rm r}=1$.