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...
Random constraint satisfaction problems play an important role in computer science and combinatorics. For example, they provide challenging benchmark instances for algorithms and they have been harnessed in probabilistic constructions of combinatorial structures with peculiar features. In an important contribution [Krzakala et al., PNAS 2007] physicists made several predictions on the precise location and nature of phase transitions in random constraint satisfaction problems. Specifically, they predicted that their satisfiability thresholds are quite generally preceded by several other thresholds that have a substantial impact both combinatorially and computationally. These include the condensation phase transition, where long-range correlations between variables emerge, and the reconstruction threshold. In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems. Additionally, we obtain contiguity results that have implications on Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g., [Banks et al., COLT 2016]).
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The jigsaw percolation process, introduced by Brummitt, Chatterjee, Dey and Sivakoff, was inspired by a group of people collectively solving a puzzle. It can also be seen as a measure of whether two graphs on a common vertex set are "jointly connected". In this paper we consider the natural generalisation of this process to an arbitrary number of graphs on the same vertex set. We prove that if these graphs are random, then the jigsaw percolation process exhibits a phase transition in terms of the product of the edge probabilities. This generalises a result of Bollobás, Riordan, Slivken and Smith.Crn(ln n) i−1 for some 2 ≤ i ≤ r or P 1 ≤ ln n Crn then whp G does not percolate;(ii) if P i ≥ Cr n(ln n) i−1 for every 2 ≤ i ≤ r and P 1 ≥ Cr ln n n , then whp G percolates.
1. The description and analysis of animal behaviour over long periods of time is one of the most important challenges in ecology. However, most of these studies are limited due to the time and cost required by human observers. The collection of data via video recordings allows observation periods to be extended. However, their evaluation by human observers is very time-consuming. Progress in automated evaluation, using suitable deep learning methods, seems to be a forwardlooking approach to analyse even large amounts of video data in an adequate time frame. 2. In this study we present amulti-step convolutional neural network system for detecting animal behaviour states, which works with high accuracy. An important aspect of our approach is the introduction of model averaging and post-processing rules to make the system robust to outliers. 3. Our trained system achieves an in-domain classification accuracy of >0.92, which is improved to >0.96 by a postprocessing step. In addition, the whole system performs even well in an out-of-domain classification task with two unknown types, achieving an average accuracy of 0.93. We provide our system at https://github.com/Klimroth/Video-Action-Classifier-for-African-Ungulates-in-Zoos/tree/main/mrcnn_based so that interested users can train their own models to classify images and conduct behavioural studies of wildlife. 4. The use of a multi-step convolutional neural network for fast and accurate classification of wildlife behaviour facilitates the evaluation of large amounts of image data in ecological studies and reduces the effort of manual analysis of images to a high degree. Our system also shows that post-processing rules are a suitable way to make species-specific adjustments and substantially increase the accuracy of the description of single behavioural phases (number, duration). The results in the out-of-domain classification strongly suggest that our system is robust and achieves a high degree of accuracy even for new species, so that other settings (e.g. field studies) can be considered.
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