Trivial, Critical and Near-critical Scaling Limits of?Two-dimensional Percolation

It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, p _c , sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to p _c . The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to p _c . The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.     

Excess 1/f noise in systems with an exponentially wide spectrum of resistances and dual universality of the percolation-like noise exponent

The excess 1/f noise in a random lattice with bond resistances r∼exp(−λx), where x is a random variable and λ≪1, is studied theoretically. It is shown that if the correlation function {δr ²}∼r r ^θ+2, then the relative spectral density of the noise in the system is expressed as C _e∼λ^m exp(−λ(1−p _c)), where p _c is the percolation threshold and m=νd (ν is the critical exponent of the correlation length and d is the dimensionality of the problem). It is hypothesized that the exponent m possesses a dual universality: It is independent of 1) the geometry of the lattice and 2) the θ-mechanism responsible for the generation of the local noise. Numerical modeling in a three-dimensional lattice gives m=52.3 for θ=1 and θ=0, in agreement with the hypothesis. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 8, 614–618 (25 April 1996)     

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE ₆ paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.Research partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).Research partially supported by the U.S. NSF under grant DMS-01-04278.     

Elementary Quantum Mechanics in a Space-Time Lattice

Studies of quantum fields and gravity suggest the existence of a minimal length, such as Planck length. It is natural to ask how the existence of a minimal length may modify the results in elementary quantum mechanics (QM) problems familiar to us. In this paper we address a simple problem from elementary non-relativistic quantum mechanics, called “particle in a box”, where the usual continuum (1+1)-space-time is supplanted by a space-time lattice. Our lattice consists of a grid of λ ₀×τ ₀ rectangles, where λ ₀, the lattice parameter, is a fundamental length (say Planck length) and, we take τ ₀ to be equal to λ ₀/c. The corresponding Schroedinger equation becomes a difference equation, the solution of which yields the q-eigenfunctions and q-eigenvalues of the energy operator as a function of λ ₀. The q-eigenfunctions form an orthonormal set and both q-eigenfunctions and q-eigenvalues reduce to continuum solutions as λ ₀→0. The corrections to eigenvalues because of the assumed lattice is shown to be O(l₀²)O(\lambda_{0}^{2}). We then compute the uncertainties in position and momentum, Δx, Δp for the box problem and study the consequent modification of Heisenberg uncertainty relation due to the assumption of space-time lattice, in contrast to modifications suggested by other investigations.     

Critical behavior of two-dimensional magnetic lattice gas model

Using Monte Carlo simulations and finite-size scaling, we investigate the critical behavior of two-dimensional magnetic lattice gas at densities ρ = 0.90, 0.95, 1.0. There is a ferromagnetic phase transition at each density. As expected, the critical temperature T _c depends on system density ρ. Unexpectedly, there is a density dependence of the critical exponent of correlation length ν. For densities ρ = 0.90,0.95,1.0, we obtain the inverse of critical exponent 1/ν = 0.835(5), 0.905(5), 1.00(1) respectively. It is found that the ratios of critical exponent β/ν and γ/ν of magnetization and susceptibility are independent of density.     

Relationships between a microscopic parameter and the stochastic equations for interface's evolution of two growth models

The relationship between a microscopic parameter p, that is related to the probability of choosing a mechanism of deposition, and the stochastic equation for the interface's evolution is studied for two different models. It is found that in one model, that is similar to ballistic deposition, the corresponding stochastic equation can be represented by a Kardar-Parisi-Zhang (KPZ) equation where both λ and ν depend on p in the following way: ν(p) = νp and λ(p) = λp ^3/2. Furthermore, in the other studied model, which is similar to random deposition with relaxation, the stochastic equation can be represented by an Edwards-Wilkinson (EW) equation where ν depends on p according to ν(p) = νp ². It is expected that these results will help to find a framework for the development of stochastic equations starting from microscopic details of growth models. Received 26 August 2002 / Received in final form 20 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: ealbano@inifta.unlp.edu.ar     

Is realization of Majorana neutrino and neutrinoless double beta decay possible in the framework of standard weak interactions?

Usually it is supposed that Majorana neutrino produced in the superposition state χ _L = ν _L + (ν _L ) ^c and then follows the neutrinoless double beta decay. But since the standard weak interactions are chiral invariant then neutrino at production has definite helicity (ν _L and (ν _L ) ^c have opposite spirality). Then these neutrinos are separately produced and their superposition state cannot appear. Thus we see that for unsuitable helicity the neutrinoless double β decay is not possible even if it is supposed that neutrino is a Majorana particle (i.e. there is not a lepton number which is conserved). Also transition of Majorana neutrino ν _L into antineutrino (ν _L ) ^c at their oscillations is forbidden since helicity in vacuum holds. Transition Majora neutrino ν _L into (ν _R ) ^c (i.e., ν _L → (ν _R ) ^c ) at oscillations is unobserved since it is supposed that mass of (ν _R ) ^c is very big. If neutrino is a Dirac particle there can be transition of ν _L neutrino into (sterile) antineutrino $ \bar v_R $ \bar v_R (i.e., ν _L → $ \bar v_R $ \bar v_R ) at neutrino oscillations if there takes place double violation of lepton number. It is necessary also to remark that introducing of a Majorana neutrino implies violation of global and local gauge invariance in the standard weak interactions.     

Conservation laws for classical and relativistic collisions. II

On the basis of elementary symmetry arguments it is shown that (1) if in classical mechanics there exists a quantity λ+Σ_iμ_iυ_i+1/2νυ ² that is conserved, where λ,μ _i, andν are particle parameters, then theμ _i andν are all proportional to a single parameterμ and the quantityAμ+Σ_iB_iμυ_i+C(λ+ 1/2Dμυ ²), whereD ≡ν/μ, is conserved for all values ofA, B _i, andC; (2) if in relativistic mechanics there exists a quantity λ+Σ_iμ_iυ_i[1−(υ ²/c ²)]^−1/2+νc[1−(υ ²/c ²)]^−1/2 that is conserved, then theμ _i andν are all proportional to a single parameterμ and the quantityAλ+Σ_iB_iμν_i[1−(υ ²/c ²)]^−1/2+Cμc [1−(υ ²/c ²)]^−1/2 is conserved for all values ofA, B _i, andC.     

Non-equilibrium phase transition of the fully frustrated square lattice Coulomb gas model

The non-equilibrium phase transitions of the fullyfrustrated (f = 1/2) square lattice Coulomb gas (CG) modeldriven by external electrical fields are studied in the frameworkof the short-time dynamic scaling approach. The criticaltemperature T_c, the static and dynamic critical exponents2β/ν, ν, and z are obtained for several smalldriving fields. The results show that T_c decreases with theincrease of electric field, and 2β/ν and z arestrongly dependent on the external electric field. Interestingly,contrary to the equilibrium case, in the presence of smallelectric field, the calculated exponent ν is close to that inpure 2D Ising model, which provides numerical evidence thatexternal electric field may change the universality class of thef = 1/2 CG system.     

Conservation laws for classical and relativistic collisions. I

On the basis of simple kinematic arguments it is shown that any quantity, depending only on the nature and velocity of a particle, that is conserved in a collision must, in classical mechanics, be of the form λ+Σ_iμ_iυ_i+1/2vυ ² or in relativistic mechanics of the form λ+Σ_iμ_iυ_i[1−(υ ²/c ²)]^−1/2+νc [1−(υ ²/c ²)]^−1/2 where λ,μ _i, andν are particle parameters.     

Scaling properties of hadron production in the Ising model for quark-hadron phase transition

Earlier study of quark-hadron phase transition in the Ginzberg-Landau theory is reexamined in the Ising model, so that spatial fluctuations during the transition can be taken into account. Although the dimension of the physical system is 2, as will be argued, bothd=2 andd=4 Ising systems are studied, the latter being theoretically closer to the Ginzberg-Landau theory. The normalized factorial momentsF _q are used to quantify multiplicity fluctuations, and the scaling exponentν is used to characterize the scaling properties. It is found by simulation on the Ising lattice thatν becomes a function of the temperatureT nearT _c . The average value ofν over a range ofT<T _c agrees with the value of 1.3 derived analytically from the Ginzberg-Landau theory. Thus the implications of the mean-field theory are not invalidated by either the introduction of spatial fluctuations or the restriction to a 2D system.     

Breit–Wigner Approximation and the Distribution¶of Resonances

For operators with a discrete spectrum, {λ _j ²}, the counting function of λ _j 's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑ _j δ_{λ j} ((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λ _j ∈ℂ₊, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that

Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents

According to recent progresses in the finite size scaling theory of disordered systems, thermodynamic observables are not self-averaging at critical points when the disorder is relevant in the Harris criterion sense. This lack of self-averageness at criticality is directly related to the distribution of pseudo-critical temperatures T_c(i,L) over the ensemble of samples (i) of size L. In this paper, we apply this analysis to disordered Poland-Scheraga models with different loop exponents c, corresponding to marginal and relevant disorder. In all cases, we numerically obtain a Gaussian histogram of pseudo-critical temperatures T_c(i,L) with mean T_c^av(L) and width ΔT_c(L). For the marginal case c=1.5 corresponding to two-dimensional wetting, both the width ΔT_c(L) and the shift [T_c(∞)-T_c^av(L)] decay as L^-1/2, so the exponent is unchanged (ν_random=2=ν_pure) but disorder is relevant and leads to non self-averaging at criticality. For relevant disorder c=1.75, the width ΔT_c(L) and the shift [T_c(∞)-T_c^av(L)] decay with the same new exponent L^-1/νrandom (where ν_random ∼2.7 > 2 > ν_pure) and there is again no self-averaging at criticality. Finally for the value c=2.15, of interest in the context of DNA denaturation, the transition is first-order in the pure case. In the presence of disorder, the width ΔT_c(L) ∼L^-1/2 dominates over the shift [T_c(∞)-T_c^av(L)] ∼L^-1, i.e. there are two correlation length exponents ν=2 and that govern respectively the averaged/typical loop distribution.     

Perturbative and non-perturbative Kolmogorov turbulence in a gluon plasma

In numerical simulations of nonabelian plasma instabilities in the hard-loop approximation, a turbulent spectrum has been observed that is characterized by a phase-space density of particles n(p)∼p ^−ν with exponent ν≃2, which is larger than expected from relativistic 2↔2 scatterings. Using the approach of Zakharov, L’vov and Falkovich, we analyze possible Kolmogorov coefficients for relativistic (m≥4)-particle processes, which give at most ν=5/3 perturbatively for an energy cascade. We discuss non-perturbative scenarios which lead to larger values. As an extreme limit we find the result ν=5 generically in an inherently non-perturbative effective field theory situation, which coincides with results obtained by Berges et al. in large-N scalar field theory. If we instead assume that scaling behavior is determined by Schwinger–Dyson resummations such that the different scaling of bare and dressed vertices matters, we find that intermediate values are possible. We present one simple scenario, which would single out ν=2.     

Monte Carlo studies of percolation phenomena for a simple cubic lattice

The site-percolation problem on a simple cubic lattice is studied by the Monte Carlo method. By combining results for periodic lattices of different sizes through the use of finite-size scaling theory we obtain good estimates forp _c (0.3115±0.0005), (0.41±0.01), (1.6±0.1), and(0.8±0.1). These results are consistent with other studies. The shape of the clusters is also studied. The average surface area for clusters of sizek is found to be close to its maximal value for the low-concentration region as well as for the critical region. The percentage of particles in clusters of different sizesk is found to have an exponential tail for large values ofk forP <p _c. Forp >p _c there is too much scatter in the data to draw firm conclusions about the size distribution.Work supported in part by USAFOSR Grant #73-2430B and by ERDA #E(11-1)-3077.     

Ground state structure of some double-λ hypernuclei by a three-body model using a simple coordinate space approach

New experimental data on the binding energyB _λλ of_λλ6He, reported very recently, come up with the valuesB _λλ = 725 ±0.14 MeV and ΔB_λλ = 101 ±0.2 MeV which are substantially lower than the old dataB _λλ = 109 ±0.8 MeV and ΔB_λλ = 4.7±10 MeV in use in literature since 1966. In view of the new data we decided to undertake a re-study of the _λλ ⁶ He hypernucleus using the same three-body model (α-λ-λ) with a simple coordinate space variational approach which was employed earlier with the old data on_λλ/6He. After fitting different λ-λ potentials to the new data of _λλ ⁶ He we have applied our method to study some double-λ hypernuclei in light, medium and heavy mass regions and have determined the structural quantities like Bλλ, the r.m.s. values of core-λ (〈r_core-λ〉〉) and λ-λ (〈r_λ-λ〉〉) distances theoretically. The core-λ interaction considered is of Woods-Saxon type. The strength and the range of the core-A potential have been adjusted to reproduce the λ-binding energy(B _λ) . These are in good agreement with the relativistic mean field (RMF) results. Our study shows that the λ-λ bonding energy ΔB_λλ decreases with increasing mass number from _λλ ¹⁰ Be to _λλ ²¹⁰ Pb of a double-A hypernucleus     

One-dimensional Anderson model with dichotomic correlation in the presence of external electric field

A one-dimensional diagonal tight binding electronic system with dichotomic correlated disorder in the presence of external d.c field is investigated. It is found numerically that the conductance distribution obeys fairly well to log-normal distribution in weak disorder strength in localized regime, which indicates validity of single parameter scaling theory in this limit. Contrary to the universal cumulant relation C ₁ = 2C ₂ in the absence of d.c. field, we demonstrated numerically that C ₁ ≫ 2C ₂ in the presence of the field in localized regime. We interpret this result as suppression of the fluctuation effects by the external field. In addition, it is obtained that the quantity NF _c , here N is the system size and F _c is the crossover field, decreases as the as the system energy E increases. Moreover, we find numerically a simple linear relation between the average logarithm of the conductance 〈ln(g)〉 and the field strength as 〈ln(g)〉 = C(N, λ)F, here C(N, λ) is a constant for particular values of N and λ, which is the Poisson parameter of the dichotomic process.     

Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on ℤ ₊ ^d where to each edgee is associated a random variable v(e). A polymer configuration is represented by a directed pathr and has a weight exp[-β ∑ _e ∈r ν(e)], withβ=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations     

Elastic fields and physical properties of surface quantum dots

Elastic fields in a system consisting of a surface coherent axisymmetric quantum dot-island on a massive substrate have been theoretically studied using the finite element method. An analysis of the influence of the quantum dot shape (form factor) and relative size (aspect ratio) δ on the accompanying elastic fields has revealed two critical quantum dot dimensions, δ _c1 and δ _c2. For δ > δ _c1, the fields are independent of the quantum dot shape and aspect ratio. At δ ≥ δ _c2, the quantum dot top remains almost undistorted. Variation of the stress tensor component σ _zz (z is the quantum dot axis of symmetry) reveals a region of tensile stresses, which is located in the substrate under the quantum dot at a particular distance from the interface. Using an approximate analytical formula for the radial component of displacements, model electron microscopy images have been calculated for quantum dot islands with δ > δ _c1 in the InSb/InAs system. The possibility of stress relaxation occurring in the system via the formation of a prismatic interstitial dislocation loop has been considered.