Moderate Deviations for Random Field Curie-Weiss Models

The random field Curie-Weiss model is derived from the classical Curie-Weiss model by replacing the deterministic global magnetic field by random local magnetic fields. This opens up a new and interestingly rich phase structure. In this setting, we derive moderate deviations principles for the random total magnetization S _n , which is the partial sum of (dependent) spins. A typical result is that under appropriate assumptions on the distribution of the local external fields there exist a real number m, a positive real number ??, and a positive integer k such that (S _n ?nm)/n ^?? satisfies a moderate deviations principle with speed n ^1?2k(1???) and rate function ??x ^2k /(2k)!, where 1?1/(2(2k?1))<??<1.     

Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians

We consider the integrated density of states (IDS) ρ_∞(λ) of random Hamiltonian H_ω=?Δ+V_ω, V_ω being a random field on ? ^d which satisfies a mixing condition. We prove that the probability of large fluctuations of the finite volume IDS |Λ|^?1ρ(λ, H_Λ(ω)), Λ ? ? ^d , around the thermodynamic limit ρ_∞(λ) is bounded from above by exp {?k|Λ|},k>0. In this case ρ_∞(λ) can be recovered from a variational principle. Furthermore we show the existence of a Lifshitztype of singularity of ρ_∞(λ) as λ → 0⁺ in the case where V_ω is non-negative. More precisely we prove the following bound: ρ_∞(λ)≦exp(?kλ^?d/2) as λ → 0⁺ k>0. This last result is then discussed in some examples.     

Conducting to non-conducting transition in dual transmission lines using a ternary model with long-range correlated disorder

In this work we study the behavior of the allowed and forbidden frequencies in disordered classical dual transmission lines when the values of capacitances {Cj} are distributed according to a ternary model with long-range correlated disorder. We introduce the disorder from a random sequence with a power spectrum S(k)∝k−(2α−1), where α?0.5 is the correlation exponent. From this sequence we generate an asymmetric ternary map using two map parameters b₁ and b₂, which adjust the occupancy probability of each possible value of the capacitances Cj={CA,CB,CC,}. If the sequence of capacitance values is totally at random α=0.5 (white noise), the electrical transmission line is in the non-conducting state for every frequency ω. When we introduce long-range correlations in the distribution of capacitances, the electrical transmission lines can change their conducting properties and we can find a transition from the non-conducting to conducting state for a fixed system size. This implies the existence of critical values of the map parameters for each correlation exponent α. By performing finite-size scaling we obtain the asymptotic value of the map parameters in the thermodynamic limit for any α. With these data we obtain a phase diagram for the symmetric ternary model, which separates the non-conducting state from the conducting one. This is the fundamental result of this Letter. In addition, introducing one or more impurities in random places of the long-range correlated distribution of capacitances, we observe a dramatic change in the conducting properties of the electrical transmission lines, in such a way that the system jumps from conducting to non-conducting states. We think that this behavior can be considered as a possible mechanism to secure communication.     

Signs of the Ising model Ursell functions

It is proven that the Ursell functionsU _2k of the Ising model have the conjectured signs: (?1) ^k+1 U _2k ≧0. The proof is based on Aizenman's random current representation and combinatorics.     

Critical behavior of long linear k-mers on honeycomb lattices

Monte Carlo (MC) simulations, finite-size scaling and theoretical analysis have been carried out to study the critical behavior of long linear particles of length k (k-mers) on honeycomb lattices. A nematic phase, characterized by a big domain of parallel k-mers, is separated from the isotropic state, by a continuous transition occurring at a finite density θ_c. Our study allowed: (1) to determine the minimum value of k (k_min), which allows the formation of the nematic phase, being k_min=11; (2) to predict the dependence of θ_c on k, being θ_c(k)∝k⁻¹; and (3) to obtain the critical exponents, which indicate that the transition belongs to the 2D three-state Potts universality class.     

Magnetization curves of randomly oriented ferromagnetic single-domain nanoparticles with combined symmetry of magnetic anisotropy

The magnetization curves of randomly oriented nanoparticles with combined symmetry of magnetic anisotropy were studied. The composite mode of the Stoner–Wolfarth model has been used. In terms of this model each nanoparticle is characterized by random cubic crystalline magnetic anisotropy and by random uniaxial magnetic anisotropy. The series of simulated magnetization curves have been obtained. Each curve corresponds to different contributions of cubic and uniaxial magnetic anisotropy energy to the full energy of an individual nanoparticle k_u. Within this series we discuss the values of remnant magnetization, coercive force, both initial and maximal susceptibilities as the function of k_u. It is found that the magnetic properties are not monotonous functions of k_u. We discuss the possibility of comparing the calculated magnetization curves with the experimental curves in order to obtain new information on the magnetic constant.     

Structural optimization of HfTiSiO high-k gate dielectrics by utilizing in-situ PVD-based fabrication method

We investigated the optimum structure for Ti-containing Hf-based high-k gate dielectrics to achieve EOT scaling below 1 nm. TiO₂/HfSiO/SiO₂ trilayer and HfTiSiO/SiO₂ bilayer structures were fabricated by a newly developed in-situ PVD-based method. We found that thermal diffusion of Ti atoms to SiO₂ underlayers degrades the EOT-J_g characteristics. Our results clearly demonstrated the impact of the trilayered structure with TiO₂ capping for improving EOT-J_g characteristics of the gate stack. We achieved an EOT scaling of 0.78 nm as well as reduced gate leakage of 7.2 × 10⁻² A/cm² for a TiO₂/HfSiO/SiO₂ trilayered high-k dielectric while maintaining the electrical properties at the bottom interface.     

Logarithmic correlation functions in two-dimensional turbulence

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the presence of three-dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory c_8,1 describes the 2D-turbulence both in the absence and in the presence of the perturbation. We obtain the energy spectrum E(k) ∼ k^−5.125 ln(k) for perturbed 2D-turbulence and E(k) ∼ k⁻⁵ ln(k) for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.     

Random pseudofractal networks with competition

In this paper, we present a simple rule which assigns fitness to each edge to generate random pseudofractal networks (RPNs). This RPN model is both scale-free and small-world. We obtain the theoretical results that the power-law exponent is γ=2+1/(1+α) for the tunable parameter α>-1, and that the degree distribution is of an exponential form for others. Analytical results also show that an RPN has a large clustering coefficient and can process hierarchical structure as C(k)∼k^-1 that is in accordance with many real networks. And we prove that the mean distance L(N) scales slower logarithmically with network size N. In particular, we explain the effect of nodes with degree 2 on the clustering coefficient. These results agree with numerical simulations very well.     

A new deterministic complex network model with hierarchical structure

We introduce a new simple pseudo tree-like network model, deterministic complex network (DCN). The proposed DCN model may simulate the hierarchical structure nature of real networks appropriately and have the unique property of ‘skipping the levels’, which is ubiquitous in social networks. Our results indicate that the DCN model has a rather small average path length and large clustering coefficient, leading to the small-world effect. Strikingly, our DCN model obeys a discrete power-law degree distribution P(k)∝k^−γ, with exponent γ approaching 1.0. We also discover that the relationship between the clustering coefficient and degree follows the scaling law C(k)∼k⁻¹, which quantitatively determines the DCN's hierarchical structure.     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

A scale-free network with limiting on vertices

We propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity. The random graph process is defined as follows: at any time-step t, (i) with probability α(k) and independently of other time-step, each vertex is inactive which means it cannot be connected by more edges, where k is the degree of v_i at the time-step t; (ii) a new vertex v_t is added along with m edges incident with v_t at one time and its neighbors are chosen in the manner of preferential attachment. We prove that the degree distribution P(k) of this random graph process satisfies if α(⋅) is a constant α₀; and P(k)∝C₂k⁻³ if α(?)↓0 as ?↑∞, where C₁,C₂ are two positive constants. The analytical result is found to be in good agreement with that obtained by numerical simulations. Furthermore, we get the degree distributions in this model with m-varying functions by simulation.     

The magnitude of the parton intrinsic transverse momentum

We analyze data on the transverse momentum distribution of dimuons produced by 22 GeV and 16 GeV π^? incident on a Cu target. We evaluate the quantum chromodynamic lowest order perturbation contributions and also introduce an intrinsic transverse momentum k_T on the partons relative to their parent hadrons. The analysis of our data leads to 〈k_T〉=0.30?0.35 GeV. We discuss our result in comparison with analyses of data at higher energies.     

Zero point energies in light cone field theory

In this paper, we study the role of zero point energies in light front quantized field theories using a simple scalar field model with quartic coupling. In the equal time formalism, the zero point energies are renormalized by normal ordering with respect to some vacuum state, which is varied to determine the true, interacting vacuum. On the light front, we shall see that this procedure acquires an unexpected subtlety due to the equivalence of the ultraviolet (k ⁺→∞) and infrared (k ⁺→0) limits of the light front momentum. In order for the divergent zero point contributions toP ⁺ andP ^? to cancel, we find that the product of the infrared and ultraviolet cutoffs must be a finite constant whose value is determined by the coupling constants of the theory. As an application, we determine the vacuum structure of the theory in two dimensions as a function of the quartic coupling. Finally, we discuss the implications of our result for the discretized versions of light front quantization.     

Growing networks with preferential deletion and addition of edges

A preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed. At each time step t=1,2,…, with probability π₁>0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree, with probability π₂ a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex, and with probability π₃=1−π₁−π₂<1/2 a vertex is chosen proportionally to its degree and a random edge of this vertex is deleted. The model is intended to capture a situation where high-degree vertices are more dynamic than low-degree vertices in the sense that their connections tend to be changing. A recursion formula is derived for the expected asymptotic fraction p_k of vertices with degree k, and solving this recursion reveals that, for π₃<1/3, we have p_k∼k^{−(3−7π₃)/(1−3π₃)}, while, for π₃>1/3, the fraction p_k decays exponentially at rate (π₁+π₂)/2π₃. There is hence a non-trivial upper bound for how much deletion the network can incorporate without losing the power-law behavior of the degree distribution. The analytical results are supported by simulations.     

The effective thermal conductivity of porous media based on statistical self-similarity

A fractal model is presented based on the thermal-electrical analogy technique and statistical self-similarity of fractal saturated porous media. A dimensionless effective thermal conductivity of saturated fractal porous media is studied by the relationship between the dimensionless effective thermal conductivity and the geometrical parameters of porous media with no empirical constant. Through this study, it is shown that the dimensionless effective thermal conductivity decreases with the increase of porosity (?) and pore area fractal dimension (Df) when ks/kg>1. The opposite trends is observed when ks/kg<1. In addition, the dimensionless effective thermal conductivity decreases with increasing tortuous fractal dimension (Dt). The model predictions are compared with existing experimental data and the results show that they are in good agreement with existing experimental data.     

The Coulomb correction to electron pair production by intermediate-energy photons

The Coulomb correction to the total cross section for intermediate-energy pair production is evaluated in terms of a high-energy-expansion formula, with the Davies-Bethe-Maximon correction as the leading term, followed by terms proportional to k^?1ln²k, k^?1lnk, k^?1 etc., the coefficients of which are determined by fitting to the Øverb?-Mork-Olsen exact low-energy results. The formula is estimated to be accurate within a few tenths of a per cent for photon energies between 3.5 m_ec² and infinity.     

On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles

We removem-balls of centersw ₁,...,w _m with the same radius α/m from a bounded domain Ω inR ³ with smooth boundary γ. Let μ _k (α/m;w(m)) denote thek-th eigenvalue of the Laplacian in Ω/m-balls under the Dirichlet condition. We consider μ _k (α/m;w(m)) as a random variable on a probability space (w ₁,...,w _m)∈Ω × ... × Ω and we examine a precise behaviour of μ _k (α/m;w(m)) asm → ∞. We give an elaboration of. M. Kac's theorem.     

Negative Binomial Distribution and the multiplicity moments at the LHC

In this work we show that the latest LHC data on multiplicity moments C₂-C₅ are well described by a two-step model in the form of a convolution of the Poisson distribution with energy-dependent source function. For the source function we take Γ Negative Binomial Distribution. No unexpected behavior of Negative Binomial Distribution parameter k is found. We give also predictions for the higher energies of 10 and 14 TeV.