Spin Glass Field Theory with Replica Fourier Transforms

We develop a field theory for spin glasses using Replica Fourier Transforms (RFT). We present the formalism for the case of replica symmetry and the case of replica symmetry breaking on an ultrametric tree, with the number of replicas $n$ and the number of replica symmetry breaking steps $R$ generic integers. We show how the RFT applied to the two-replica fields allows to construct a new basis which block-diagonalizes the four-replica mass-matrix, into the replicon, anomalous and longitudinal modes. The eigenvalues are given in terms of the mass RFT and the propagators in the RFT space are obtained by inversion of the block-diagonal matrix. The formalism allows to express any $i$-replica vertex in the new RFT basis and hence enables to perform a standard perturbation expansion. We apply the formalism to calculate the contribution of the Gaussian fluctuations around the Parisi solution for the free-energy of an Ising spin glass.Comment: 39 pages, 3 figure

Statistical mechanics of the random K-SAT model

The Random K-Satisfiability Problem, consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M=alpha N random logical clauses containing K variables each, is studied using the replica symmetric framework of diluted disordered systems. We present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest K=2, K>= 3 and K>>1. The calculation of the number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to predict a first order jump at the threshold where the Boolean expressions become unsatisfiable with probability one, is thoroughly displayed. In the case K=2, the (rigorously known) critical value (alpha=1) of the number of clauses per Boolean variable is recovered while for K>=3 we show that the system exhibits a replica symmetry breaking transition. The annealed approximation is proven to be exact for large K.Comment: 34 pages + 1 table + 8 fig., submitted to Phys. Rev. E, new section added and references update

Is the droplet theory for the Ising spin glass inconsistent with replica field theory?

Symmetry arguments are used to derive a set of exact identities between irreducible vertex functions for the replica symmetric field theory of the Ising spin glass in zero magnetic field. Their range of applicability spans from mean field to short ranged systems in physical dimensions. The replica symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8 and d<6 the resummation of an infinite number of terms is necessary to settle the problem. When d<8, these Ward-like identities must be used to distinguish an Almeida-Thouless line from the replica symmetric droplet phase.Comment: 4 pages. Accepted for publication in J.Phys.A. This is the accepted version with the following minor changes: one extra sentence in the abstract; footnote 2 slightly extended; last paragraph somewhat reformulate

Finite dimensional corrections to mean field in a short-range p-spin glassy model

In this work we discuss a short range version of the $p$-spin model. The model is provided with a parameter that allows to control the crossover with the mean field behaviour. We detect a discrepancy between the perturbative approach and numerical simulation. We attribute it to non-perturbative effects due to the finite probability that each particular realization of the disorder allows for the formation of regions where the system is less frustrated and locally freezes at a higher temperature.Comment: 18 pages, 5 figures, submitted to Phys Rev

Quenched Random Graphs

Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.Comment: 9 pages, report CPTH-A264.109

On the hybrid origin of Narcissus biflorus (Amaryllidaceae): analysis of C-banding and rDNA structure

Abstract Giemsa and fluorochrome banding with DAPI and chromomycin A3, were utilized to assess karyological details which correlate N. biflorus with the parental species: N. tazetta and N. poeticus. The banding profile in N. biflorus clearly reproduced the model of its progenitors. The EcoR1 restriction pattern of rDNA obtained by Southern blot hybridization indicated, in our material, that each species has more than one ribosomal gene type and in N. biflorus both the ribosomal repeat units of the progenitor species are present

Sherrington-Kirkpatrick model near T=TcT=T_c: expanding around the Replica Symmetric Solution

An expansion for the free energy functional of the Sherrington-Kirkpatrick (SK) model, around the Replica Symmetric SK solution $Q^{({\rm RS})}_{ab} = \delta_{ab} + q(1-\delta_{ab})$ is investigated. In particular, when the expansion is truncated to fourth order in. $Q_{ab} - Q^{({\rm RS})}_{ab}$. The Full Replica Symmetry Broken (FRSB) solution is explicitly found but it turns out to exist only in the range of temperature $0.549...\leq T\leq T_c=1$, not including T=0. On the other hand an expansion around the paramagnetic solution $Q^{({\rm PM})}_{ab} = \delta_{ab}$ up to fourth order yields a FRSB solution that exists in a limited temperature range $0.915...\leq T \leq T_c=1$.Comment: 18 pages, 3 figure

MEDSLIK-II, a Lagrangian marine surface oil spill model for short-term forecasting – Part 2: Numerical simulations and validations

Abstract. In this paper we use MEDSLIK-II, a Lagrangian marine surface oil spill model described in Part 1 (De Dominicis et al., 2013), to simulate oil slick transport and transformation processes for realistic oceanic cases, where satellite or drifting buoys data are available for verification. The model is coupled with operational oceanographic currents, atmospheric analyses winds and remote sensing data for initialization. The sensitivity of the oil spill simulations to several model parameterizations is analyzed and the results are validated using surface drifters, SAR (synthetic aperture radar) and optical satellite images in different regions of the Mediterranean Sea. It is found that the forecast skill of Lagrangian trajectories largely depends on the accuracy of the Eulerian ocean currents: the operational models give useful estimates of currents, but high-frequency (hourly) and high-spatial resolution is required, and the Stokes drift velocity has to be added, especially in coastal areas. From a numerical point of view, it is found that a realistic oil concentration reconstruction is obtained using an oil tracer grid resolution of about 100 m, with at least 100 000 Lagrangian particles. Moreover, sensitivity experiments to uncertain model parameters show that the knowledge of oil type and slick thickness are, among all the others, key model parameters affecting the simulation results. Considering acceptable for the simulated trajectories a maximum spatial error of the order of three times the horizontal resolution of the Eulerian ocean currents, the predictability skill for particle trajectories is from 1 to 2.5 days depending on the specific current regime. This suggests that re-initialization of the simulations is required every day

Stability of self-consistent solutions for the Hubbard model at intermediate and strong coupling

We present a general framework how to investigate stability of solutions within a single self-consistent renormalization scheme being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we impose equations of motion on the one- as well on the two-particle Green functions simultaneously and introduce approximations in their input, the completely irreducible two-particle vertex. Thereby we do not loose singularities caused by multiple two-particle scatterings. We find a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density type or anomalous. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices get unstable before a long-range order or a metal-insulator transition can be reached. We use the parquet approximation and turn it to a workable scheme with dynamical vertex corrections. We derive a qualitatively new theory with two-particle self-consistence, the complexity of which is comparable with FLEX-type approximations. We show that it is the simplest consistent and stable theory being able to describe qualitatively correctly quantum critical points and the transition from weak to strong coupling in correlated electron systems.Comment: REVTeX, 26 pages, 12 PS figure

Using retinex for point selection in 3D shape registration

Inspired by retinex theory, we propose a novel method for selecting key points from a depth map of a 3D freeform shape; we also use these key points as a basis for shape registration. To find key points, first, depths are transformed using the Hotelling method and normalized to reduce their dependence on a particular viewpoint. Adaptive smoothing is then applied using weights which decrease with spatial gradient and local inhomogeneity; this preserves local features such as edges and corners while ensuring smoothed depths are not reduced. Key points are those with locally maximal depths, faithfully capturing shape. We show how such key points can be used in an efficient registration process, using two state-of-the-art iterative closest point variants. A comparative study with leading alternatives, using real range images, shows that our approach provides informative, expressive, and repeatable points leading to the most accurate registration results. © 2014 Elsevier Ltd