Scale-invariance of random populations: From Paretian to Poissonian fractality

Random populations represented by stochastically scattered collections of real-valued points are abundant across many fields of science. Fractality, in the context of random populations, is conventionally associated with a Paretian distribution of the population's values.Using a Poissonian approach to the modeling of random populations, we introduce a definition of “Poissonian fractality” based on the notion of scale-invariance. This definition leads to the characterization of four different classes of Fractal Poissonian Populations—three of which being non-Paretian objects. The Fractal Poissonian Populations characterized turn out to be the unique fixed points of natural renormalizations, and turn out to be intimately related to Extreme Value distributions and to Lévy Stable distributions.     

Testing nucleation theory in two dimensions

We calculate bubble-nucleation rates for (2+1)-dimensional scalar theories at high temperature. Our approach is based on the notion of a real coarse-grained potential. The region of applicability of our method is determined through internal consistency criteria. We compare our results with data from lattice simulations. Good agreement is observed when the renormalized action of the simulated theory is known.     

Dynamics of selfavoiding tethered membranes. II. Inclusion of hydrodynamic interaction (Zimm model)

The dynamical scaling properties of selfavoiding polymerized membranes with internal dimension D embedded into d dimensions are studied including hydrodynamical interactions. It is shown that the theory is renormalizable to all orders in perturbation theory and that the dynamical scaling exponent z is given by z=d. The crossover to the region, where the membrane is crumpled swollen but the hydrodynamic interaction irrelevant is discussed. The results apply as well to polymers (D=1) as to membranes (D=2). Received: 5 September 1997 / Accepted: 17 November 1997     

Statistical behaviors for renormalization of correlated permeability field

In this article, the statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated. In contrast to the uncorrelated porous media, the scaling of the variance of the renormalized permeability field exhibits a crossover behavior. When the correlation lengths are larger compared with the domain scale covered by the renormalization procedure, the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero. As the renormalization number increases, the covered domain scale will eventually become larger than the correlation lengths, and then the scaling property will transit to the uncorrelated case. The convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated. Both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction. For the log-normal permeability field, numerical simulations show that the crossover scaling and also the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions.     

Position space renormalization group study of the spin-1 random semi-infinite Blume–Capel model

We study the spin-1 Blume–Capel model under a random crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal–Kadanoff technique. Interesting results are obtained, which tell us that the randomness destroys the first order phase transitions and only those of the second order occur. We give the list of nine fixed points and their topology describing the surface critical behavior. Five new types of phase diagram are found with a rich variety of phase transitions, in accordance with the values of the bulk and surface probabilities and the ratios linking bulk and surface interactions.     

Loop variables and gauge invariant exact renormalization group equations for (open) string theory II

In arXiv:1202.4298 gauge invariant interacting equations were written down for the spin 2 and spin 3 massive modes using the exact renormalization group of a world sheet theory. This is generalized to all the higher levels in this paper. An interacting theory of an infinite tower of massive higher spins is obtained. They appear as a compactification of a massless theory in one higher dimension. The compactification and consequent mass is essential for writing the interaction terms. Just as for spin 2 and spin 3, the interactions are in terms of gauge invariant “field strengths” and the gauge transformations are the same as for the free theory. This theory can then be truncated in a gauge invariant way by removing one oscillator of the extra dimension to match the field content of BRST string (field) theory. The truncation has to be done level by level and results are given explicitly for level 4. At least up to level 5, the truncation can be done in a way that preserves the higher-dimensional structure. There is a relatively straightforward generalization of this construction to (arbitrary) curved space–time and this is also outlined.     

Renormalization group approach to a p-wave superconducting model

We present in this work an exact renormalization group (RG) treatment of a one-dimensional p-wave superconductor. The model proposed by Kitaev consists of a chain of spinless fermions with a p-wave gap. It is a paradigmatic model of great actual interest since it presents a weak pairing superconducting phase that has Majorana fermions at the ends of the chain. Those are predicted to be useful for quantum computation. The RG allows to obtain the phase diagram of the model and to study the quantum phase transition from the weak to the strong pairing phase. It yields the attractors of these phases and the critical exponents of the weak to strong pairing transition. We show that the weak pairing phase of the model is governed by a chaotic attractor being non-trivial from both its topological and RG properties. In the strong pairing phase the RG flow is towards a conventional strong coupling fixed point. Finally, we propose an alternative way for obtaining p-wave superconductivity in a one-dimensional system without spin–orbit interaction.     

Renormalization in Coulomb gauge QCD

In the Coulomb gauge of QCD, the Hamiltonian contains a non-linear Christ–Lee term, which may alternatively be derived from a careful treatment of ambiguous Feynman integrals at 2-loop order. We investigate how and if UV divergences from higher order graphs can be consistently absorbed by renormalization of the Christ–Lee term. We find that they cannot.     

Quantum gravitational contributions to the beta function of quantum electrodynamics

We show in a diagrammatic and regularization independent analysis that the quadratic contribution to the beta function which has been conjectured to render quantum electrodynamics asymptotically free near the Planck scale has its origin in a surface term. Such surface term is intrinsically arbitrarily valued and it is argued to vanish in a consistent treatment of the model.