Biased measures for random Constraint Satisfaction Problems: larger interaction range and asymptotic expansion

TL;DR

This paper studies how non-uniform weighting and extended interactions affect the clustering transition in random hypergraph bicoloring problems, revealing increased thresholds and asymptotic behaviors.

Contribution

It introduces a generalized model with larger interaction ranges and derives an asymptotic expansion for the clustering threshold in the large $k$ limit.

Findings

The clustering threshold $ ext{α}_d(k)$ increases with non-uniform weights.

An asymptotic formula for $ ext{α}_d(k)$ is derived for large $k$.

The constant $ ext{γ}_d$ exceeds that of the uniform measure.

Abstract

We investigate the clustering transition undergone by an exemplary random constraint satisfaction problem, the bicoloring of $k$-uniform random hypergraphs, when its solutions are weighted non-uniformly, with a soft interaction between variables belonging to distinct hyperedges. We show that the threshold $\alpha_{\rm d}(k)$ for the transition can be further increased with respect to a restricted interaction within the hyperedges, and perform an asymptotic expansion of $\alpha_{\rm d}(k)$ in the large $k$ limit. We find that $\alpha_{\rm d}(k) = \frac{2^{k-1}}{k}(\ln k + \ln \ln k + \gamma_{\rm d} + o(1))$, where the constant $\gamma_{\rm d}$ is strictly larger than for the uniform measure over solutions.