Generalized Central Limit Theorem and Renormalization Group

We introduce a simple instance of the renormalization group transformation in the Banach space of probability densities. By changing the scaling of the renormalized variables we obtain, as fixed points of the transformation, the Lévy strictly stable laws. We also investigate the behavior of the transformation around these fixed points and the domain of attraction for different values of the scaling parameter. The physical interest of a renormalization group approach to the generalized central limit theorem is discussed.     

Renormalization Group Study of Ising Transition in Double－Frequency sine－Gordon Model

We utilize the renormallzation group (RG) technique to analyze the Ising critical behavior in the double frequency Sine-Gordon model. The one-loop RG equations obtained show unambiguously that there exist two Ising critical points besides the trivial Gaussian fixed point. The topology of the RG flows is obtained as well.     

Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume–Emery–Griffiths Models

Renormalization group based on the Migdal–Kadanoff bond removal approach is often considered a simple and valuable tool to understand the critical behavior of complicated statistical mechanical models. In presence of quenched disorder, however, predictions obtained with the Migdal–Kadanoff bond removal approach quite often fail to quantitatively and qualitatively reproduce critical properties obtained in the mean-field approximation or by numerical simulations in finite dimensions. In an attempt to overcome this limitation we analyze the behavior of Ising and Blume–Emery–Griffiths models on more structured hierarchical lattices. We find that, apart from some exceptions, the failure is not limited to Midgal–Kadanoff cells but originates right from the hierarchization of Bravais lattices on small cells, and shows up also when in-cell loops are considered.     

Phase Transition in Ferromagnetic Ising Models with Non-uniform External Magnetic Fields

In this article we study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field. We show that the Ising model on the hypercubic lattice with a summable magnetic field has a first-order phase transition and, for any positive (resp. negative) and bounded magnetic field, the model does not present the phase transition phenomenon whenever lim?inf?h _i >0, where \(\mathbf{h}=(h_{i})_{i\in \mathbb{Z}^{d}}\) is the external magnetic field.     

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice ℤ3

We consider an Euclidean supersymmetric field theory in ℤ³ given by a supersymmetric Φ⁴ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ³. The Green’s function depends on the Lévy-Khintchine parameter with 0<α<2. For the Φ⁴ interaction is marginal. We prove for sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ³ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ³.     

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. Received: 29 January 2001 / Accepted: 8 March 2001     

Gaussian Random Matrix Models¶for q-deformed Gaussian Variables

We construct a family of random matrix models for the q-deformed Gaussian random variables G _μ=a _μ+a^*_μ, where the annihilation operators a _μ and creation operators $a\gwia_\nu$ fulfill the $q$-deformed commutation relation a _μ a^*_ν−q a^*_ν a _μ=Γ_μν, Γ_μν is the covariance and 0<q<1 is a given number. An important feature of the considered random matrices is that the joint distribution of their entries is Gaussian. Received: 29 March 2000 / Accepted: 1 August 2000     

Extremal Properties for Weakly Correlated Random Variables Arising in Speckle Patterns

Extremal properties of the statistics of speckle pattern are studied in the context of so-called “optically smoothed” light beams of laser-matter interaction. It is shown that the asymptotic statistics of the highest intensity in a speckle pattern, which can be associated with the most intense speckles, follows a Gumbel law, which is in agreement with numerical simulations. It is found that the probability density function of the most intense speckle peaks around the value corresponding to the logarithm of the number of speckles in the considered volume times the average intensity value of the speckle pattern. This result is of great interest for nonlinear processes, like instabilities, where extreme speckles play an important role.     

Luttinger Revisited—The Renormalization Group Approach

Luttinger's contributions abound in different parts of many-body physics. Here I review the ones that appear when one uses the Renormalization Group (RG) to study the subject: the Luttinger Liquid, Luttinger's Theorem (on the volume of the Fermi surface) and the Kohn–Luttinger Theorem on the superconducting instability of all metals as one approaches absolute zero.     

Correlation Decay and Recurrence Asymptotics for Some Robust Nonuniformly Hyperbolic Maps

We study a robust class of multidimensional non-uniformly hyperbolic transformations considered by Oliveira and Viana (Ergod. Theory Dyn. Syst. 28:501–533, 2008). For an open class of Hölder continuous potentials with small variation we show that the unique equilibrium state has exponential decay of correlations and that the distribution of hitting times is asymptotically exponential. Furthermore, using that the equilibrium states satisfy a weak Gibbs property we also prove log-normal fluctuations of the return times around their average.     

Open and Closed World Models in Kaluza-Klein-Theory with Variables G and Λ

The field equation of higher dimensions theory, have been applied in the area of cosmology. The resulting differential equations are solved for open and closed. We derive a relation between the Einstein constant G(t) and the cosmological constant Λ(t) from the conservation law T _μ ^ν _;ν =0. We give a specific form of Λ(t) to solve the non-linear differential equations. Some cosmological parameters are calculated and some relevant cosmological problems are discussed.     

Limiting Shapes for Deterministic Centrally Seeded Growth Models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in ℤ ^d , Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter h (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension d≥1. For the rotor router model, the limiting shape is a sphere for all values of h. For one of the sandpile models, and h=2d−2 (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit h→−∞. Finally, we prove that the rotor router shape contains a diamond.     

Application of Corner Transfer Matrix Renormalization Group Method to the Correlation Function of a Two－Dimensional Ising Model ＊

The correlation function of a two-dimensional Ising model is calculated by the corner transfer matrix renormal-ization group method. We obtain the critical exponent η= 0.2496 with few computer resources.     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

Coexistence of Ordered and Disordered Phases in Potts Models in the Continuum

This is the second of two papers on a continuum version of the Potts model, where particles are points in ℝ ^d , d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ ⁻¹, γ>0. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist S+1 mutually distinct DLR measures.     

Entanglement Dynamics of Ising Model with Dzialoshinskii-Moriya Interaction in an Inhomogeneous Magnetic Field

This paper investigates the entanglement dynamics of two-spin Ising model with the Dzialoshinskii-Moriya (DM) interaction in an inhomogeneous magnetic field. The system is initially prepared in the Werner state.It is found that the phenomenon of the entanglement sudden death (ESD) appears during the evolution process. We also analyze in detail the effects of the DM interaction, the degree of inhomogeneous magnetic field and the purity of the initial state on the time of ESD. These effects indicate that one can control the entanglement of the system. It is helpful for quantum information processing.     

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q ⁻⁴⁷ (resp. q ⁻⁴⁶). Finally, we compute chromatic roots for strips of widths 9≤m≤12 with free boundary conditions and locate roughly the limiting curves.     

Effect Algebras Are Not Adequate Models for Quantum Mechanics

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states.     

Information-Geometric Indicators of Chaos in Gaussian Models on Statistical Manifolds of Negative Ricci Curvature

A new information-geometric approach to chaotic dynamics on curved statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical manifold ℳ _s underlying an ED Gaussian model describing an arbitrary system of 3N degrees of freedom leads to linear information-geometric entropy growth and to exponential divergence of the Jacobi vector field intensity, quantum and classical features of chaos respectively.     

Bi-partite Entanglement Entropy in Massive QFT with a Boundary: the Ising Model

In this paper we give an exact infinite-series expression for the bi-partite entanglement entropy of the quantum Ising model in the ordered regime, both with a boundary magnetic field and in infinite volume. This generalizes and extends previous results involving the present authors for the bi-partite entanglement entropy of integrable quantum field theories, which exploited the generalization of the form factor program to branch-point twist fields. In the boundary case, we isolate in a universal way the part of the entanglement entropy which is related to the boundary entropy introduced by Affleck and Ludwig, and explain how this relation should hold in more general QFT models. We provide several consistency checks for the validity of our form factor results, notably, the identification of the leading ultraviolet behaviour both of the entanglement entropy and of the two-point function of twist fields in the bulk theory, to a great degree of precision by including up to 500 form factor contributions.