Effective field theory for models defined over small-world networks. First and second order phase transitions

We present an effective field theory method to analyze, in a very general way, models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it provides, yielding a clear and immediate (also in terms of calculation) physical insight, the exact critical behavior and the exact critical surfaces and percolation thresholds. The underlying structure of the non random part of the model, i.e., the set of spins filling up a given lattice L_0 of dimension d_0 and interacting through a fixed coupling J_0, is exactly taken into account. When J_0\geq 0, the small-world effect gives rise, as is known, to a second-order phase transition that takes place independently of the dimension d_0 and of the added random connectivity c. When J_0<0, a different and novel scenario emerges in which, besides a spin glass transition, multiple first- and second-order phase transitions may take place. As immediate analytical applications we analyze the Viana-Bray model (d_0=0), the one dimensional chain (d_0=1), and the spherical model for arbitrary d_0.Comment: 28 pages, 18 figures; merged version of the manuscripts arXiv:0801.3454 and arXiv:0801.3563 conform to the published versio

Laser pulse analysis

Methods are presented for locating threshold points by using laser pulse analysis. It was found that there are errors involved in the determination of each of these quantities, and an attempt was made to separate their effects on the overall range correction. Several series of corrected range measurements for fixed reflectors and satellites were obtained. Residuals were computed by fitting the range measurements to either fixed-reflector distances or short arcs of satellite orbits. Root mean square values of these residuals are presented

Geometry, stochastic calculus and quantum fields in a non-commutative space-time

The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the non-commutativity of the space-time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space-time, on quantum stochastic calculus and on the construction of quantum fields. Some of these effects are studied in this paper.Comment: 36 pages Latex, 1 eps figur

Invariant measures for Cherry flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.Comment: 12 pages; updated versio

Stratification of the orbit space in gauge theories. The role of nongeneric strata

Gauge theory is a theory with constraints and, for that reason, the space of physical states is not a manifold but a stratified space (orbifold) with singularities. The classification of strata for smooth (and generalized) connections is reviewed as well as the formulation of the physical space as the zero set of a momentum map. Several important features of nongeneric strata are discussed and new results are presented suggesting an important role for these strata as concentrators of the measure in ground state functionals and as a source of multiple structures in low-lying excitations.Comment: 22 pages Latex, 1 figur

Diluted antiferromagnet in a ferromagnetic enviroment

The question of robustness of a network under random ``attacks'' is treated in the framework of critical phenomena. The persistence of spontaneous magnetization of a ferromagnetic system to the random inclusion of antiferromagnetic interactions is investigated. After examing the static properties of the quenched version (in respect to the random antiferromagnetic interactions) of the model, the persistence of the magnetization is analysed also in the annealed approximation, and the difference in the results are discussed