Scaling Properties of the Probability Distribution of Lattice Gribov Copies

We study the problem of the Landau gauge fixing in the case of the SU(2) lattice gauge theory. We show that the probability to find a lattice Gribov copy increases considerably when the physical size of the lattice exceeds some critical value $\approx2.75/\sqrt{\sigma}$, almost independent of the lattice spacing. The impact of the choice of the copy on Green functions is presented. We confirm that the ghost propagator depends on the choice of the copy, this dependence decreasing for increasing volumes above the critical one. The gluon propagator as well as the gluonic three-point functions are insensitive to choice of the copy (within present statistical errors). Finally we show that gauge copies which have the same value of the minimisation functional ($\int d^4x (A^a_\mu)^2$) are equivalent, up to a global gauge transformation, and yield the same Green functions.Comment: replaced with revised version; 23 pages, 7 figures, 27 table

Non-Perturbative Approach to the Landau Gauge Gluodynamics

We discuss a non-perturbative lattice calculation of the ghost and gluon propagators in the pure Yang-Mills theory in Landau gauge. The ultraviolet behaviour is checked up to NNNLO yielding the value \Lambda^{n_f=0}_{\ms}=269(5)^{+12}_{-9}\text{MeV}, and we show that lattice Green functions satisfy the complete Schwinger-Dyson equation for the ghost propagator for all considered momenta. The study of the above propagators at small momenta showed that the infrared divergence of the ghost propagator is enhanced, whereas the gluon propagator seem to remain finite and non-zero. The result for the ghost propagator is consistent with the analysis of the Slavnov-Taylor identity, whereas, according to this analysis, the gluon propagator should diverge in the infrared, a result at odds with other approaches.Comment: To appear in the proceedings of the workshop "Hadron Structure and QCD: from LOW to HIGH energies" (St. Petersburg, Russia, 20-24 September 2005

Asymptotic behavior of the ghost propagator in SU3 lattice gauge theory

We study the asymptotic behavior of the ghost propagator in the quenched SU(3) lattice gauge theory with Wilson action. The study is performed on lattices with a physical volume fixed around 1.6 fm and different lattice spacings: 0.100 fm, 0.070 fm and 0.055 fm. We implement an efficient algorithm for computing the Faddeev-Popov operator on the lattice. We are able to extrapolate the lattice data for the ghost propagator towards the continuum and to show that the extrapolated data on each lattice can be described up to four-loop perturbation theory from 2.0 GeV to 6.0 GeV. The three-loop values are consistent with those extracted from previous perturbative studies of the gluon propagator. However the effective \Lambda_{\ms} scale which reproduces the data does depend strongly upon the order of perturbation theory and on the renormalization scheme used in the parametrization. We show how the truncation of the perturbative series can account for the magnitude of the dependency in this energy range. The contribution of non-perturbative corrections will be discussed elsewhere.Comment: 26 pages, 7 figure

The Infrared Behaviour of the Pure Yang-Mills Green Functions

We study the infrared behaviour of the pure Yang-Mills correlators using relations that are well defined in the non-perturbative domain. These are the Slavnov-Taylor identity for three-gluon vertex and the Schwinger-Dyson equation for ghost propagator in the Landau gauge. We also use several inputs from lattice simulations. We show that lattice data are in serious conflict with a widely spread analytical relation between the gluon and ghost infrared critical exponents. We conjecture that this is explained by a singular behaviour of the ghost-ghost-gluon vertex function in the infrared. We show that, anyhow, this discrepancy is not due to some lattice artefact since lattice Green functions satisfy the ghost propagator Schwinger-Dyson equation. We also report on a puzzle concerning the infrared gluon propagator: lattice data seem to favor a constant non vanishing zero momentum gluon propagator, while the Slavnov-Taylor identity (complemented with some regularity hypothesis of scalar functions) implies that it should diverge.Comment: 25 pages, 7 figures; replaced version with some references adde and an enlarged discussion of the non-renormalization theorem; second replacement with improved figures and added reference

Is the QCD ghost dressing function finite at zero momentum ?

We show that a finite non-vanishing ghost dressing function at zero momentum satisfies the scaling properties of the ghost propagator Schwinger-Dyson equation. This kind of Schwinger-Dyson solutions may well agree with lattice data and provides an interesting alternative to the widely spread claim that the gluon dressing function behaves like the inverse squared ghost dressing function, a claim which is at odds with lattice data. We demonstrate that, if the ghost dressing function is less singular than any power of $p$, it must be finite non-vanishing at zero momentum: any logarithmic behaviour is for instance excluded. We add some remarks about coupled Schwinger-Dyson analyses.Comment: 8 pages, 2 figure

Programmable Quantum Annealers as Noisy Gibbs Samplers

Drawing independent samples from high-dimensional probability distributions represents the major computational bottleneck for modern algorithms, including powerful machine learning frameworks such as deep learning. The quest for discovering larger families of distributions for which sampling can be efficiently realized has inspired an exploration beyond established computing methods and turning to novel physical devices that leverage the principles of quantum computation. Quantum annealing embodies a promising computational paradigm that is intimately related to the complexity of energy landscapes in Gibbs distributions, which relate the probabilities of system states to the energies of these states. Here, we study the sampling properties of physical realizations of quantum annealers which are implemented through programmable lattices of superconducting flux qubits. Comprehensive statistical analysis of the data produced by these quantum machines shows that quantum annealers behave as samplers that generate independent configurations from low-temperature noisy Gibbs distributions. We show that the structure of the output distribution probes the intrinsic physical properties of the quantum device such as effective temperature of individual qubits and magnitude of local qubit noise, which result in a non-linear response function and spurious interactions that are absent in the hardware implementation. We anticipate that our methodology will find widespread use in characterization of future generations of quantum annealers and other emerging analog computing devices.Comment: 6 pages, 4 figures, with 36 pages of Supplementary Informatio

Constraints on the IR behaviour of gluon and ghost propagator from Ward-Slavnov-Taylor identities

We consider the constraints of the Slavnov-Taylor identity of the IR behaviour of gluon and ghost propagators and their compatibility with solutions of the ghost Dyson-Schwinger equation and with the lattice picture.Comment: 5 pages, 2 figure

Epidemic spreading and bond percolation on multilayer networks

The Susceptible-Infected-Recovered (SIR) model is studied in multilayer networks with arbitrary number of links across the layers. By following the mapping to bond percolation we give the analytical expression for the epidemic threshold and the fraction of the infected individuals in arbitrary number of layers. These results provide an exact prediction of the epidemic threshold for infinite locally tree-like multilayer networks, and an lower bound of the epidemic threshold for more general multilayer networks. The case of a multilayer network formed by two interconnected networks is specifically studied as a function of the degree distribution within and across the layers. We show that the epidemic threshold strongly depends on the degree correlations of the multilayer structure. Finally we relate our results to the results obtained in the annealed approximation for the Susceptible-Infected-Susceptible (SIS) model.Comment: 8 pages, 2 figure