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Register a new userCriticality of tuning in athermal phase transitions
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We experimentally address the importance of tuning in athermal phase transitions, which are triggered only by a slowly varying external field acting as tuning parameter. Using higher order statistics of fluctuations, a singular critical instability is detected for the first time in spite of an apparent universal self-similar kinetics over a broad range of driving force. The results as well as the experimental technique are likely to be of significance to many slowly driven non-equilibrium systems from geophysics to material science which display avalanche dynamics.
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The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed. We focus on the character of criticality as well as on underlying symmetries and topologies that are crucial for understanding phase diagrams and the critical behavior.
Conduction through materials crucially depends on how ordered they are. Periodically ordered systems exhibit extended Bloch waves that generate metallic bands, whereas disorder is known to limit conduction and localize the motion of particles in a medium. In this context, quasiperiodic systems, which are neither periodic nor disordered, reveal exotic conduction properties, self-similar wavefunctions, and critical phenomena. Here, we explore the localization properties of waves in a novel family of quasiperiodic chains obtained when continuously interpolating between two paradigmatic limits: the Aubry-Andre model, famous for its metal-to-insulator transition, and the Fibonacci chain, known for its critical nature. Using both theoretical analysis and experiments on cavity-polariton devices, we discover that the Aubry-Andre model evolves into criticality through a cascade of band-selective localization/delocalization transitions that iteratively shape the self-similar critical wavefunctions of the Fibonacci chain. Our findings offer (i) a unique new insight into understanding the criticality of quasiperiodic chains, (ii) a controllable knob by which to engineer band-selective pass filters, and (iii) a versatile experimental platform with which to further study the interplay of many-body interactions and dissipation in a wide range of quasiperiodic models.
We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Renyi and regular random graphs and determine their asymptotic values for large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.
Many inference problems, notably the stochastic block model (SBM) that generates a random graph with a hidden community structure, undergo phase transitions as a function of the signal-to-noise ratio, and can exhibit hard phases in which optimal inference is information-theoretically possible but computationally challenging. In this paper we refine this description by emphasizing the existence of more generic phase diagrams with a hybrid-hard phase in which it is computationally easy to reach a non-trivial inference accuracy, but computationally hard to match the information theoretically optimal one. We support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs. These expansions shed light on the existence of hybrid-hard phases, for a large class of planted constraint satisfaction problems, and on the question of the tightness of the Kesten-Stigum (KS) bound for the associated tree reconstruction problem. Our results show that the instability of the trivial fixed point is not a generic evidence for the Bayes-optimality of the message passing algorithms. We clarify in particular the status of the symmetric SBM with 4 communities and of the tree reconstruction of the associated Potts model: in the assortative (ferromagnetic) case the KS bound is always tight, whereas in the disassortative (antiferromagnetic) case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the KS bound is tight and a low degree regime where it is not. We also investigate the SBM with 2 communities of different sizes, a.k.a. the asymmetric Ising model, and describe quantitatively its computational gap as a function of its asymmetry, and a version of the SBM with 2 groups of communities. We complement this study with numerical simulations of the Belief Propagation algorithm.
We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes $K$ variables that must be assigned one out of $q$ colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs ($K=2$) and bicoloring of random hypergraphs ($q=2$), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size $q$ of the domain of the variables and the arity $K$ of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem ($q=2$) where we find that for $K=3$ and $K=4$ the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q=2$, $K ge 4$. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where $q$ and/or $K$ diverge.
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