Biased measures for random constraint satisfaction problems: larger interaction range and asymptotic expansion

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 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 d(k) in the large k limit. We find that αd(k)=2k−1k(lnk+lnlnk+γd+o(1)) , where the constant γ d is strictly larger than for the uniform measure over solutions.