Biased measures for random Constraint Satisfaction Problems: larger interaction range and asymptotic expansion
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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 α_ d(k) = 2^k-1/k(ln k + lnln k + γ_ d + o(1)), where the constant γ_ d is strictly larger than for the uniform measure over solutions.
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