In 1979, G. Parisi [14] predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand [19] and later generalized to the mixed p-spin models by Panchenko [12]. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi

Structural and photoluminescence ͑PL͒ properties of InN dots grown on GaN by metal organic vapor phase epitaxy using the flow-rate modulation technique, and their dependence on growth conditions, were investigated. An ammonia ͑NH 3 ͒ background flow was intentionally supplied during indium deposition periods to control the kinetics of adatoms and hence the morphology of InN dots. Samples prepared under lower NH 3 background flows generally exhibit narrower and more intense PL signals peaked at lower emission energies. The authors point out that the NH 3 background flow is an important parameter that controls not only the nucleation process but also the emission property of InN dots.

In this paper, we define and compute the generalized TAP correction for the energy of the mixed p-spin models with Ising spins, and give the corresponding generalized TAP representation for the free energy. We study the generalized TAP states, which are the maximizers of the generalized TAP free energy, and show that their order parameter matches the order parameter of the ancestor states in the Parisi ansatz. We compute the critical point equations of the TAP free energy that generalize the classical TAP equations for pure states. Furthermore, we give an exact description of the region where the generalized TAP correction is replica symmetric, in which case it coincides with the classical TAP correction, and show that Plefka's condition is necessary for this to happen. In particular, the generalized TAP correction is not always replica symmetric on the points corresponding to the Edwards-Anderson parameter.

We show that in random K-uniform hypergraphs of constant average degree, for even K ≥ 4, local algorithms defined as factors of i.i.d. can not find nearly maximal cuts when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain non-trivial interval -a phenomenon referred to as the overlap gap property -which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models, and showing the overlap gap property in the latter setting. This paper considers the problem of algorithmically finding nearly optimal spin configurations in the diluted K-spin model. We specifically focus on local algorithms defined as factors of i.i.d., the formal definition of which is provided in Section 2. The diluted K-spin model is also known as the max-cut problem for Kuniform Erdős-Rényi hypergraphs of constant average degree, and also as the random K-XORSAT model. The problem is only interesting for even K and we prove that, for even K ≥ 4, local algorithms fail to find the nearly optimal spin configurations (maximal cuts) once the average degree is large enough.The proof is based on finding a structural constraint for the overlap of any two nearly optimal spin configurations -the overlap gap property -that goes against certain properties of local algorithms. For K = 2, the overlap gap property is not expected to hold, which is why this case is excluded. The structural constraint is derived from recent results on the mean field K-spin spin glass models, in particular, the Parisi formula and the Guerra-Talagrand replica symmetry breaking bound at zero temperature. We begin with a discussion of the model and the notion of algorithms that we use.The K-spin model The set of ±1 spin configurations on N vertices will be denoted by

We consider the problem of disorder chaos in the spherical mean-field model. It is concerned about the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra's replica symmetry breaking scheme, we establish this at the level of the free energy as well as the Gibbs measure irrespective the presence or absence of the external field.