Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

We study the stationary distribution of the standard Abelian sandpile model in the box Λ_n = [-n, n] ^d ∩ ℤ ^d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ ^d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ ^d . In the case d > 4, we also make use of Wilson’s method. An erratum to this article is available at .     

The Infinite Volume Limit of Dissipative Abelian Sandpiles

We construct the thermodynamic limit of the stationary measures of the Bak-Tang-Wiesenfeld sandpile model with a dissipative toppling matrix (sand grains may disappear at each toppling). We prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic Markov process leaving it invariant. We show how to extend the Dhar formalism of the abelian group of toppling operators to infinite volume in order to obtain a compact abelian group with a unique Haar measure representing the uniform distribution over the recurrent configurations that create finite avalanchesWork partially supported by Tournesol project – nr. T2001.11/03016VF     

A renormalization group analysis of correlation functions for the dipole gas

We develop a renormalization group method for analyzing the generating functional for charge correlations of a dilute classical dipole gas. It is based on and extends the renormalization group analysis introduced by Brydges and Yau for the dipole gas partition function. Our method leads to systematic formulas for the large-distance behavior of correlation functions of all orders. We prove that in any dimensiond2, at any value>0 of the inverse temperature, and at sufficiently small activityz, the correlation functions exhibit at large distances the same behavior as for a vacuum (z=0), but with a new dielectric constant 1+ over which we have good control. The results proved here extend existing results on the two-point correlations to all higher correlations, and constitute a general confirmation of the fact that dipoles do not screen.     

The tensor renormalization group for pure and disordered two-dimensional lattice systems

The tensor renormalization group (TRG) is a powerful new approach for coarse-graining classical two-dimensional (2D) lattice Hamiltonians. It uses the intuitive framework of traditional position space renormalization group methods-analyzing flows in the space of Hamiltonian parameters-but can be systematically improved to yield thermodynamic properties at much higher precision. We present initial results demonstrating that the TRG can be generalized to quenched random systems, applying it to obtain the phase diagram of a bond-diluted triangular lattice Ising ferromagnet. This opens a variety of potential future applications, most prominently spin glasses.     

Renormalization group approach for the site-bond percolation in structured stochastic environments

The quenched averaged percolation problem of a lattice with a given structure is analyzed. The structure is described by the static structure factorS(q)q ^–ain the regionq 0. As a result of the renormalization group, it follows that the critical behavior fora < 2 is the same as in the random percolation. In the case ofa=2 second universality class with=0 and=1/2+/8+ ²/32 is predicted.     

Bimetric truncations for quantum Einstein gravity and asymptotic safety

In the average action approach to the quantization of gravity the fundamental requirement of “background independence” is met by actually introducing a background metric but leaving it completely arbitrary. The associated Wilsonian renormalization group defines a coarse graining flow on a theory space of functionals which, besides the dynamical metric, depend explicitly on the background metric. All solutions to the truncated flow equations known to date have a trivial background field dependence only, namely via the classical gauge fixing term. In this paper, we analyze a number of conceptual issues related to the bimetric character of the gravitational average action and explore a first nontrivial bimetric truncation in the simplified setting of conformally reduced gravity. Possible implications for the Asymptotic Safety program and the cosmological constant problem are discussed in detail.     

Dipole defects in BeO

Thermally Stimulated Depolarization Currents (TSDC) technique was applied to study the effects of thermal treatments and gamma irradiation in the formation, aggregation and destruction processes of dipole defects in beryllium oxide (BeO). In this work, a TSDC dipole peak was observed at 350 K that was assigned to impurity-vacancy complexes. The TSDC peak presents a linear dependence with the Polarization Voltage (V_P), typical behaviour of defects with dipole origin. Thermal treatments of 1 h were performed between 400 and 900 °C and a decay in the peak intensity was observed. After the gamma irradiation of the thermal treated samples a restoration tendency of the TSDC peak was observed.     

On the universal behavior of two-dimensional Ising spin glasses

Ising spin glasses are studied, at zero temperature, on a hierarchical lattice as an approach to the square lattice. The stiffness exponent y, which governs the behavior of the interactions under changes of scale, is computed for several distinct continuous symmetric probability distributions for the couplings. All distributions considered lead to the same estimates, i.e., the exponent y is universal. Our results are compared with other estimates available for the two-dimensional Gaussian Ising spin glass.     

Generalized renormalization group treatment of the growth kinetics of unstable systems

We discuss the generalization of a renormalization group technique developed previously for the study of ordering in unstable systems in the context of the ferromagnetic Ising model with spin-flip dynamics. Difficulties encountered in earlier work are eliminated through the use of new recursion relations dependent on a continuous spatial rescaling factorb 1. A more careful analysis and implementation of the approximation scheme is carried out. Our improved method allows the study of the anisotropy of the time-dependent structure factor and the pre-scaling behavior of the shape function.     

Using the Renormalization Group to Classify Boolean Functions

This paper argues that the renormalization group technique used to characterize phase transitions in condensed matter systems can be used to classify Boolean functions. A renormalization group transformation is presented that maps an arbitrary Boolean function of N Boolean variables to one of N−1 variables. Applying this transformation to a generic Boolean function (one whose output for each input is chosen randomly and independently to be one or zero with equal probability) yields another generic Boolean function. Moreover, applying the transformation to some other functions known to be non-generic, such as Boolean functions that can be written as polynomials of degree ξ with ξ ≪ N and functions that depend on composite variables such as the arithmetic sum of the inputs, yields non-generic results. One can thus define different phases of Boolean functions as classes of functions with different types of behavior upon repeated application of the renormalization transformation. Possible relationships between different phases of Boolean functions and computational complexity classes studied in computer science are discussed.     

Dipole solitons in nonlocal nonlinear media with anisotropy

We investigate the existence and stability of dipole-mode solitons in two-dimensional models of nonlocal media with anisotropic Kerr nonlinearity analytically and numerically. We obtain the approximate solution of such elliptic dipole solitons by using the variational approximation. The dynamics of such dipole-mode solitons is governed by the eccentricity of both the input beam and the nonlocal response function. We also compute the stability of the solitons by direct numerical simulations. The effects of the anisotropy of the nonlocal response function on the propagation of the dipole beam are also discussed in detail.     

Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation

We review and extend in several directions recent results on the “asymptotic safety” approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. We begin by discussing various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so-called Einstein-Hilbert truncation, where only the cosmological constant and Newton’s constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one-loop divergences. We then apply the Renormalization Group to higher derivative gravity. In the case of a general action quadratic in curvature we recover, within certain approximations, the known asymptotic freedom of the four-derivative terms, while Newton’s constant and of the cosmological constant have a nontrivial fixed point. In the case of actions that are polynomials in the scalar curvature of degree up to eight we find that the theory has a fixed point with three UV-attractive directions, so that the requirement of having a continuum limit constrains the couplings to lie in a three-dimensional subspace, whose equation is explicitly given. We emphasize throughout the difference between scheme-dependent and scheme-independent results, and provide several examples of the fact that only dimensionless couplings can have “universal” behavior.     

An extended approach for computing the critical properties in the two-and three-dimensional lattices within the effective-field renormalization group method

In this letter we employing the effective-field renormalization group (EFRG) to study the Ising model with nearest neighbors to obtain the reduced critical temperature and exponents ν for bi- and three-dimensional lattices by increasing cluster scheme by extending recent works. The technique follows up the same strategy of the mean field renormalization group (MFRG) by introducing an alternative way for constructing classical effective-field equations of state takes on rigorous Ising spin identities.     

New renormalization procedure for eliminating redundant operators

A generalized exact renormalization group equation is obtained by using a new rencrmalization procedure. This equation does not contain redundant operators and therefore enables one to avoid using an uncertain procedure for their exclusion.     

Variational Principle and Almost Quasilocality for Renormalized Measures

We study the variational principle for some non-Gibbsian measures. We give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities (so-called second part of the variational principle). Applying this to noisy decimations of the low-temperature phases of the Ising model, we obtain almost sure quasilocality for these measures and the second part of the variational principle. For the projection of low temperature Ising phases on a one-dimensional layer, we also obtain the second part of the variational principle.     

Fluctuation-dissipation theorem and the dynamical renormalization group

The relation between a recently introduced dynamical real-space renormalization group and the fluctuation-dissipation theorem is discussed. An apparent incompatibility is pointed out and resolved.     

Dipole solitons in an optical lattice with longitudinal modulation

In this paper, we numerically demonstrate the (1+1)-dimensional dipole solitons can exist in a new Kerr-type optical lattice with longitudinal modulation that fades away and boosts up alternately. The solitons whose two dipoles simultaneously located at one lattice site and at two adjacent lattice sites are investigated, respectively. The results show that, in the two cases, the dipole solitons can be stably trapped in this kind of lattice by properly adjusting lattice parameters and soliton parameters when the repulsive force of dipoles balances the centripetal force resulting from the lattice potential effect on dipole solitons. In addition, the trapping of dipole solitons with an incident angle or the initial center position is discussed.