Uniform Approximation of the Integrated Density of States for Long-Range Percolation Hamiltonians

In this paper we study the spectrum of long-range percolation graphs. The underlying geometry is given in terms of a finitely generated amenable group. We prove that the integrated density of states (IDS) or spectral distribution function can be approximated uniformly in the energy variable. This result is already new for percolation on ℤ ^d . Using this, we are able to characterize the set of discontinuities of the IDS.     

Iterated Fully Coordinated Percolation on a Square Lattice

We study, on a square lattice, an extension to fully coordinated percolation which we call iterated fully coordinated percolation. In fully coordinated percolation, sites become occupied if all four of its nearest neighbors are also occupied. Repeating this site selection process again yields the iterated fully coordinated percolation model. Our results show a large enhancement in the size of highly connected regions after each iteration (from ordinary to fully coordinated and then to iterated fully coordinated percolation); enhancements that are much larger than an extension of correlations by an extra lattice constant might suggest. We also study the universality among the three problems by determining the corresponding static and dynamic critical exponents. Specifically, a new method to directly calculate the walk dimension, d _w , using finite size scaling applied to normal mode analysis is used. This method is applicable to any geometry and requires significantly less computation than previously known calculations to determine d _w .     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

Mean-Field Behavior for Long- and Finite Range Ising Model,Percolation and Self-Avoiding Walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d>2(α ∧2) for self-avoiding walk and the Ising model, and d>3(α ∧2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (Ann. Probab. 33(5):1886–1944, 2005).      

Quantum geometry from 2 + 1 AdS quantum gravity on the torus

Wilson observables for 2 + 1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick’s formula for the area of a polygon with integer vertices.     

Search for the NN → 6q phase transition in the np-polarized measurements at T kin = 1–6 GeV

The measurements of np-spin observables at 0° have been performed for the first time on the Delta-Sigma experimental facility of LHE JINR up to P _n = 4.5 GeV/c using the monochromatic neutron beam. They include detailed measurements of the Δσ _L(np) spin differences and the study of the np → pn elastic charge-exchange process. In the Δσ _L(np) and −Δσ _L(I = 0) energy dependencies over the energy region T_kin = 1.2–3.7 GeV the peculiarity at 1.8 GeV was observed. Such energy behavior was predicted by the QCD approach as a signal of the NN → 6q phase transition. For the exhaustive investigation of this effect it is necessary to measure the energy dependence of the complete set of np observables with both longitudinal (L) and transverse (T) polarizations of the neutron beam and proton target. This will allow Direct Reconstruction of all three NN forward Scattering Amplitudes (DRSA) to be performed, and the observed peculiarity to be checked around T_kin = 1.8 GeV and at the higher energies using the Argand diagrams method.     

On the Ornstein-Zernike Behaviour for the Bernoulli Bond Percolation on $\pmb{\mathbb{Z}}^{d},d\geq3$, in the Supercritical Regime

We prove Ornstein-Zernike behaviour in every direction for finite connection functions of bond percolation on ℤ ^d for d≥3 when p, the probability of occupation of a bond, is sufficiently close to 1. Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.     

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 .     

Effects of size and surface anisotropy on thermal magnetization and hysteresis in the magnetic clusters

Based on Monte Carlo simulation, the spin configurations, thermal magnetization and hysteresis loops of the clusters coated by the surface shell with radial anisotropy are studied. Interestingly, a new multidomain containing a few of subdomains whose easy directions are along those of the configurational anisotropy, a magnetization curve in steps and a first order phase transition from the single domain to the multidomain in the thermal and field magnetization processes, are found, which is as a result of the interplay of the configurational anisotropy, the size effect, the surface anisotropy, the applied field and the thermal fluctuation. In this first order transition, we find a critical temperature, a critical surface anisotropy and a critical size. The simulated temperature dependence of the coercivity of the cluster with the surface anisotropy can be fitted by H_c (T)=H_c (0)(1-C_αT^α) with low value of α, which explains well the experimental results of the nanoparticles. Moreover, it is found that the hysteresis loops and coercivity are strongly affected by the cluster size and the thickness of the surface layer.     

Domain structure of cellulose triacetate

The topology of the surface of thin films obtained from cellulose triacetate solutions has been studied using electron microscopy. The domain structure of the physical network of macromolecules (percolation cluster) with the maximum size L ≈ 30 nm of regions with a local orientational order has been established. With an increase in the volume fraction Ω of the cluster to the critical value Ω = 0.91, repacking of domains takes place, which is connected with a decrease in the parameter Ω, L doubling bifurcation of the size L, and formation of regions of long-range orientational order: the mesophase of the polymer.     

Crystal-field splitting of Pr<Superscript>3+</Superscript> ion multiplets in YPO<Subscript>4</Subscript> with allowance for interconfigurational interaction

The crystal-field splitting of the Pr³⁺ ion multiplets is analyzed with regard to the influence of excited opposite-parity configurations 4f ^{(N − 1)}5d and a configuration with charge transfer. Such an approach makes it possible to (i) refine the characterization of the Stark structure of multiplets by 39% compared with the approximation of a weak configurational interaction and (ii) determine the covalence parameters and the parameters of an odd-symmetry crystal field from experimental data for the Stark structure. The covalence parameters thus determined coincide in order of magnitude with the respective parameters calculated for other ligands using microscopic models.     

Single-ion approach to the interpretation of the x-ray photoelectron spectra of the valence bands of monoxides of 3d elements

The single-ion approach taking account of only the intra-atomic Coulomb interaction in free d ⁿ ions and in d ⁿ ions in an O _h crystal field is used to analyze the fine structure of the x-ray photoelectron spectra of the valence bands of monoxides of 3d elements. The x-ray photoelectron spectra were studied as a collection of d ⁿ⁻¹ and d ⁿ L multiplets representing the unscreened and screened parts, respectively, of the final state. The unscreened part of the final state can be described by the distribution of the line strengths of the photoelectronic transition d ⁿ → d ⁿ⁻¹ and the screened part can be described as a partially relaxed distribution of the statistical weights of the d ⁿ ions. Fiz. Tverd. Tela (St. Petersburg) 39, 1056–1063 (June 1997)     

The corrections to scaling within Mazenko’s theory in the limit of low and high dimensions

We consider corrections to scaling within an approximate theory developed by Mazenko for nonconserved order parameter in the limit of low (d → 1) and high (d → ∞) dimensions. The corrections to scaling considered here follows from the departures of the initial condition from the scaling morphology. Including corrections to scaling, the equal time correlation function has the form: C(r, t) = f ₀(r/L)+L ^−ω f ₁(r/L)+…, where L is a characteristic length scale (i.e. domain size). The correction-to-scaling exponent ω and the correction-to-scaling functions f ₁(x) are calculated for both low and high dimensions. In both dimensions the value of ω is found to be ω = 4 similar to 1D Glauber model and OJK theory (the theory developed by Ohta, Jasnow and Kawasaki).     

Θ+-Pentaquark parity from associative baryon production in πd collisions, π ± + d → Θ+ + Σ±

We derive, in model-independent way, the spin structure of the matrix element for the reaction of associative Θ⁺-pentaquark production, π ^± + d → Θ⁺ + Σ^±, in the threshold region and in collinear kinematics. The expressions for the polarization observables in this reaction are found assuming spin 1/2 and different parities for Θ⁺. We have proved that such a reaction can be used for a model-independent determination of the P parity of Θ⁺ only by measuring the Θ⁺ polarization. Other polarization observables, such as the dependence of the Σ^± polarization on the vector and tensor deuteron polarizations, are insensitive to the Θ⁺ parity under the considered kinematical conditions. The linear and quadratic relations between polarization observables in (Θ⁺ is unpolarized) do not depend on the parity of the Θ⁺ pentaquark. The analysis performed for this reaction is also applicable without any modification to the reaction K ⁺ + d → Θ⁺ + p. Using the pole model for the reaction mechanism, we estimated the value of the cross section for the reaction K ⁺ + d → Θ⁺ + p near the threshold. The text was submitted by the authors in English.     

Polarization observables in the reaction pn↦dφ

The reaction pn↦dφ is studied within a covariant boson exchange model. The behavior of polarization observables being accessible in forthcoming experiments near threshold is predicted. Received: 13 February 2001 / Accepted: 25 March 2002     

The Donaldson–Witten Function for Gauge Groups of Rank Larger Than One

We study correlation functions in topologically twisted , d=4 supersymmetric Yang–Mills theory for gauge groups of rank larger than one on compact four-manifolds X. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of b _2,+(X)>0 in terms of Seiberg–Witten invariants and the classical cohomology ring of X. For manifolds X of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over =1 vacua. We describe two applications of our expressions, one to superconformal field theory and one to large N expansions of SU(N) , d=4 supersymmetric Yang–Mills theory. Received: 30 March 1998 / Accepted: 17 April 1998     

Entropic Repulsion for the High Dimensional Gaussian Lattice Field Between Two Walls

The main aim of this paper is to discuss the entropic repulsion of random interfaces between two hard walls. We consider the d (≥ 3)-dimensional Gaussian lattice field on ℝ^λ _N , λ _N = [−N, N] ^d ∩ ℤ ^d and identify the repulsion of the field as N → ∞ under the condition that the field lies between two hard walls at the height level 0 and L in Λ _N where L is large enough but finite. We also study the same problem for two layered interfaces case.     

Quantum Stochastic Integrals as Operators

We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand—a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of a finite algebra we allow the integrator to be an L ²-martingale in which case the integrals are L ²-martingales too.