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 $alpha_{rm d}(k)$ 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.