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

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 $α_{\rm d}(k)$ for the transition can be further increased with respect to a restricted interaction within the hyperedges, and perform an asymptotic expansion of $α_{\rm d}(k)$ in the large $k$ limit. We find that $α_{\rm d}(k) = \frac{2^{k-1}}{k}(\ln k + \ln \ln k + γ_{\rm d} + o(1))$, where the constant $γ_{\rm d}$ is strictly larger than for the uniform measure over solutions.