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Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous 'cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007]. In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].1 k h=1 ρ π h (σ h ) . 2 and d cond (k, β) = inf{d > 0 : sup π∈P 2 * ({1,−1}) B k−spin (d, β, π) > ln 2}. Then 0 < d cond (k, β) < ∞ and (k, β).From now on we assume that k ≥ 4 is even. The regime d < d cond (k, β) is called the replica symmetric phase. According to the cavity method, its key feature is that with probability tending to 1 in the limit n → ∞, two independent samples σ 1 , σ 2 ('replicas') chosen from the Gibbs measure µ H,J ,β are "essentially perpendicular". To formalize this define for σ, τ : V n → {±1} the overlap as ̺ σ,τ = x∈V n σ(x)τ(x)/n. We write 〈 · 〉 H,J ,β for the average on σ 1 , σ 2 chosen independently from µ H,J ,β and denote the expectation over the choice of H and J by E [ · ]. Theorem 1.2. For all β > 0 and k ≥ 4 even we have d cond (k, β) The corresponding statement for k = 2 was proved by Guerra and Toninelli, but as they point out their argument does not extend to larger k [37]. Theorem 1.2 implies the absence of extensive long-range correlations in the replica symmetric phase. Indeed, for two vertices x, y ∈ V n and s, t ∈ {+1, −1} letbe the joint distribution of the spins assigned to x, y. Further, letρ be the uniform distribution on {±1} × {±1}. Then the total variation distance µ H,J ,β,x,y −ρ TV is a measure of how correlated the spins of x, y are. Indeed, in the case that k is even for every x ∈ V n the Gibbs marginals satisfy µ H,J ,β,x (±1) = 〈1{σ 1 (x) = ±1}〉 H,J ,β = 1/2 because µ H,J ,β (σ) = µ H,J ,β (−σ) for every σ ∈ {−1, +1} n . Therefore, if the spins at x, y were independent, then µ H,J ,β,x,y = µ H,J ,β,x ⊗ µ H,J ,β,y =ρ. Furthermore, it is well known (e.g., [13, Section 2]) thatThus, Theorem 1.2 implies that for d < d cond (k, β), with probability tending to 1, the spins assigned to two random vertices x, y of H are asymptotically independent. By contrast, Theorem 1.2 and (1.2) show that extensive longrange dependencies occur beyond but arbitrarily close to d cond (k, β).1.3. The Potts antiferromagnet. Let q ≥ 2 be an integer, let Ω = {1, . . . , q} be a set...

, where−1 is a constant. The proof relies on a new, purely probabilistic approach.

Abstract. We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Pólya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on Pólya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps. This is the extended and revised journal version of a conference paper with the title "An unbiased pointing operator for unlabeled structures, with applications to counting and sam-