The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.

The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e., models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e., not free, not determinantal) six-vertex model, and provide evidence for the following two claims:(i) the critical fluctuations are given by a Gaussian Free Field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous, meaning that its coupling constant K becomes position-dependent, K → K(x).The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of K and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green's function of the Laplacian ∆ = ∇ · ∇ inside the critical domain, but instead, of the Green's function of a generalized Laplacian ∆ = ∇ · 1 K ∇ parametrized by the function K. Surprisingly, we also find that there is a change of regime when ∆ ≤ −1/2, with K becoming singular at one point.

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 + ln ln k + γ d + o ( 1 ) ) , where the constant γ d is strictly larger than for the uniform measure over solutions.

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