Proper homothetic maps and fixed points

Let f be a proper homothetic map of the pseudo-Riemannian manifold M and assume f has a fixed point p. If all of the eigenvalues of either f* _p or f ^-1*_p have absolute values less than unity, then M is topologically R _n and M has a flat metric. This yields three characterizations of Minkowski spacetime. In general, a homothetic map of a complete pseudo-Riemannian manifold need not have fixed points. Furthermore, an example shows the existence of a proper homothetic map with a fixed point does not imply M is flat. The scalar curvature vanishes at a fixed point, but some of the sectional curvatures may be nonzero.     

Infra-red fixed points in supersymmetry

Model independent constraints on supersymmetric models emerge when certain couplings are drawn towards their infra-red (quasi) fixed points in the course of their renormalization group evolution. The general principles are first reviewed and the conclusions for some recent studies of theories with R-parity and baryon and lepton number violations are summarized.     

Critical fixed points in discrete hydrodynamics

A discrete-cell formulation of hydrodynamics was recently introduced, which is exactly renormalizable in a certain sense: if one knows the discrete equations of motion for a certain cell size W and discrete time interval τ, one can accurately numerically calculate the equations of motion on the coarser scales 2W or 2τ. These coarsening transformations have previously been investigated for the one-dimensional diffusive system. A line of fixed points was found, parameterized by the (positive) diffusivity D'. In this paper we examine the behavior of the coarsening transformation on the D' = 0 manifold in the space of equations of motion for one-dimensional systems. We find another line of fixed points, this one parameterized by the super-Burnett coefficient D'₃. This corresponds to a Gaussian critical point. The possibility of generalizing this to non-Gaussian (Ising-like) critical points is discussed.     

Exact fixed points in discrete hydrodynamics

Recently a discrete formulation of hydrodynamics was introduced, which was shown to be exactly renormalizable in a certain sense: a procedure was given for computing the equations of motion on a coarse space and time scale from those on a finer scale. In this paper we carry out this coarsening procedure explicitly, giving exact numerical results for a one-dimensional diffusive system. The coarsening transformation is found to have a one-parameter family of nontrivial fixed points, parameterized by a diffusion parameterD. This result gives a new way of understanding why so many systems obey Fick's lawj = – D'dn/dx. Any of an extremely broad class of microscopic equations of motion, when viewed on a coarse enough scale, obey the fixed-point equations (which are equivalent to Fick's law). The methods used here are equally applicable to higher-dimensionality systems such as fluids.Research partially supported by the Chemistry Division of NSF through Grant No. CHE-7906649.     

GUT precursors and fixed points in higher-dimensional theories

Within the context of traditional logarithmic grand unification atM _GUT ≈ 10¹⁶ GeV, we show that it is nevertheless possible to observe certain GUT states such asX andY gauge bosons at lower scales, perhaps even in the TeV range. We refer to such states as ‘GUT precursors’. Such states offer an interesting alternative possibility for new physics at the TeV scale, even when the scale of gauge coupling unification remains high, and suggest that it may be possible to probe GUT physics directly even within the context of high-scale gauge coupling unification. More generally, our results also suggest that it is possible to construct self-consistent ‘hybrid’ models containing widely separated energy scales, and give rise to a Kaluza-Klein realization of non-trivial fixed points in higher-dimensional gauge theories.     

Complex networks renormalization: flows and fixed points

Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under renormalization, such as the maximum number of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems. An analysis of classic renormalization for percolation and the Ising model on the lattice confirms this analogy. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.     

Infrared behavior and fixed points in Landau-gauge QCD

We investigate the infrared behavior of gluon and ghost propagators in Landau-gauge QCD by means of an exact renormalization group equation. We explain how, in general, the infrared momentum structure of Green functions can be extracted within this approach. An optimization procedure is devised to remove residual regulator dependences. In Landau-gauge QCD this framework is used to determine the infrared leading terms of the propagators. The results support the Kugo-Ojima confinement scenario. Possible extensions are discussed.     

Causality and fixed points in the renormalization group

It is discussed under which conditions the singularity of type exp(-1/g)-predicted from causality for logarithmic models-will appear in the leading order of the effective coupling constant.     

Voronoi cell properties from simulated and real random spheres and points

Results are reported of a computer-aided study of the properties of Voronoi cells generated from randomly placed impenetrable uniform spheres which saturate a space and of Poisson points in three-dimensional space. An empirical expression is suggested for the volume distribution function and an expression for the probability of thenth nearest neighbor of a Voronoi cell nucleus forming a cell face. Comparison is made with the work of Meijering [Philips Res. Rep. 8:270 (1953)], Finney [Proc. R. Soc. (London) A319:479, 495 (1970)], Kiang [Z. Astrophys. 64:433 (1966)], and others. Radial distributions, coordination number, and packing density are discussed for the spheres.     

Homothetic transformations with fixed points in spacetime

A study is made of homothetic motions with fixed points in spacetime. Some general properties of such spacetimes are established, and a characterization of generalized plane wave spacetimes is proved. A general discussion of homothetic motions in Einstein's theory is given.This is in the sense that no open subset ofM is flat.     

Critical theory of overscreened Kondo fixed points

The low-temperature fixed point of the Kondo model, for k bands and a spin-s impurity, is well understood by Nozières' Fermi liquid theory for k 2s. However when k > 2s, a new type of non-trivial fixed point is known to occur. We study this fixed point using higher-level Kac-Moody conformal field theory and Cardy's approach to boundary critical phenomena. The specific heat and magnetization are shown to be determined by the leading irrelevant operator and the corresponding critical exponents are obtained exactly. The Wilson ratio is argued to be universal and its exact value is also calculated. The asymptotic finite-size spectrum is determined. Thermodynamic exponents agree precisely with the Bethe ansatz; for k = 2, S = 1/2, the Wilson ratio also agrees well with the approximate value obtained from the Bethe ansatz; the slope of the β-function agrees with the perturbative result in the large-k limit and the finite-size spectrum agrees excellently with approximate results obtained previously by Wilson's numerical renormalization group method in the case k = 2, S = 1/2.     

On the fixed points for circle maps

Renormalization group transformations have been developed to study the critical behavior of circle maps. When the winding number equals the golden mean, the fixed point functions must satisfy two functional equations. However, it turns out that one of these equations already determines the fixed point solutions. It is shown that under certain conditions the second functional equation is automatically satisfied.     

Renormalization group fixed points in Yukawa theories

The Callan-Symanzik functions of Yukawa theories in which the scalar mesons transform as the regular representation of SU(3), SU(2) or U(1) are calculated in two-loop order. An attractive renormalisation group fixed point away from the origin is found, but at a distance such that perturbation theory can not be considered reliable.     

Gravitational fixed points and asymptotic safety from perturbation theory

The fixed point structure of the renormalization flow in Einstein gravity and higher derivative gravity is investigated in terms of the background effective action. A refined perturbative framework is developed consisting of: use of a covariant operator regularization that keeps track of powerlike divergences, a non-minimal subtraction ansatz for the originally dimensionful couplings in combination with a ‘Wilsonian’ matching condition, and the construction of a one-loop effective action exactly gauge-independent on-shell in regularized form. Using this framework strictly positive fixed points for the dimensionless Newton constant gN${\mathrm{g}}_N$ and the cosmological constant λ   can be identified already in one-loop perturbation theory. The renormalization flow is asymptotically safe with respect to the nontrivial fixed points in both cases. In Einstein gravity a residual gauge dependence of the fixed points is unavoidable while in higher derivative gravity both the fixed point and the flow equations are universal. Along this flow spectral positivity of the Hessians can be satisfied, thereby meeting an essential condition for a well-defined Euclidean field theory setting. Dependence on O(10)$O(10)$ initial data is erased to accuracy 0.5%$0.5\text{\%}$ after O(100)$O(100)$ units of the renormalization mass scale and the flow settles on a λ(gN)$\lambda ({\mathrm{g}}_N)$ orbit.     

Quasi-Gaussian fixed points and factorial cumulants in nuclear multifragmentation

We re-analyze the conditions for the phenomenon of intermittency (self-similar fluctuations) to occur in models of multifragmentation. Analyzing two different mechanisms, the bond-percolation and the ERW (Elattari, Richert and Wagner) statistical fragmentation models, we point out a common quasi-Gaussian shape of the total multiplicity distribution in the critical range. The fixed-point property is also observed for the multiplicity of the second bin. Fluctuations are studied using scaled factorial cumulants instead of scaled factorial moments. The second-order cumulant displays the intermittency signal while higher order cumulants are equal to zero, revealing a large information redundancy in scaled factorial moments. A practical criterion is proposed to identify the Gaussian feature of light-fragment production, distinguishing between a self-similarity mechanism (ERW) and the superposition of independent sources (percolation).