Percolation cluster numbers

It has recently been suggested that there may be an infinite number of independent exponents hidden in the tails of the probability distribution of average percolation cluster numbers. A simple approximation of non-Gaussian effects was used to deduce this result and we show that this approximation is questionable. Extensive simulations of the cluster distribution have been made and an interesting dependence of the cumulants on concentration and range of summation has been observed.     

Differential real-space renormalization of thed-Dimensional Gaussian model

With the aid of the differential real-space method we derive exact renormalization group (RG) equations for the Gaussian model ind dimensions. The equations involved + 1 spatially dependent nearest-neighbor interactions. We locate a critical fixed point and obtain the exact thermal critical indexy _T = 2. A special trajectory of the full nonlinear RG transformation is found and the free energy of the corresponding initial state calculated.Supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 130 Ferroelektrika.     

Correlated percolation — a real-space renormalization group study

A quenched-correlated percolation system is studied by using the real-space renormalization group method. We chose the nearest-neighbour short-range order α as the correlation parameter. By treating the correlated probability self-consistently in terms of site occupation probability p and α, we found that, in two dimensions, there is only one nontrivial physical fixed point at random percolation and the correlation is an irrelevant parameter which always leads to the same universality.     

Anderson localization for the diamond lattice studied by a real-space renormalization

The problem of Anderson localization for strongly disordered electronic systems on a diamond lattice is studied by a real-space renormalization for a very large system of 27,000 sites. The renormalization, which is exact in principle, is based on the transformation of the system considered into an equivalent chain system. The mobility edges as a function of the strength of disorder and the critical value for the Anderson transition are calculated.     

Differential real-space renormalization of the Gaussian model in three dimensions

We show how the differential real-space renormalization method can be extended beyond two dimensions by applying it to the Gaussian model on the fcc and diamond lattice.     

Percolation, renormalization, and quantum computing with nondeterministic gates

We apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of optical architectures, where probabilistic gates are common, and cold atoms in optical lattices, where hole defects occur. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of conditional dynamics; we furthermore show that except for a single layer of gates during the preparation, all subsequent operations can be shifted to the final adapted single-qubit measurements. Remarkably, cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available.     

Percolation cluster sizes and perimeters in three dimensions

From exact perimeter polynomials of Sykes et al ind=3 dimensions we determine the average perimeter s _n of clusters, the width of the distribution about the average value, and the numberc _n of clusters containingn occupied sites each. The exponent, defined through log (c _n ) —n for largen, is found to be consistent with the predictions (p < p _c ) = 1 and (p>p _c )=(d—1)/d.     

Critical dynamics by real-space renormalisation group

The Niemeijer-van Leeuwen renormalisation group method is extended to study critical dynamics. Its simplest application to the kinetic Ising model yields the critical dynamic exponentz1.7 in two dimensions.     

Percolation and cluster distribution. II. layers,variable-range interactions,and exciton cluster model

Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 10⁶ sites. We investigate for the square lattice the variable-range percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponents and. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.Supported by NSF Grant DMR76-07832.     

Interaction-driven real-space condensation

We study real-space condensation in a broad class of stochastic mass transport models. We show that the steady state of such models has a pair-factorized form which generalizes the standard factorized steady states. The condensation in this class of models is driven by interactions which give rise to a spatially extended condensate that differs fundamentally from the previously studied examples. We present numerical results as well as a theoretical analysis of the condensation transition and show that the criterion for condensation is related to the binding-unbinding transition of solid-on-solid interfaces.     

Percolation and cluster distribution. III. Algorithms for the site-bond problem

Algorithms for estimating the percolation probabilities and cluster size distribution are given in the framework of a Monte Carlo simulation for disordered lattices for the generalized site-bond problem. The site-bond approach is useful when a percolation process cannot be exclusively described in the context of pure site or pure bond percolation. An extended multiple labeling technique (ECMLT) is introduced for the generalized problem. The ECMLT is applied to the site-bond percolation problem for square and triangular lattices. Numerical data are given for lattices containing up to 16 million sites. An application to polymer gelation is suggested.Supported by NSF Grant DMR76-07832 and NIH Grant NS08116-9.     

渗流相变实验

基于渗流相变理论设计了可重复使用的渗流实验仪器。在大学物理甚至高中物理一次实验课的时间内,利用渗流实验仪进行测量不同随机占据概率下的电流,对四次实验结果进行平均后,绘制相对电导率与占据概率曲线,可以获得与文献[9]相似的结果,再现渗流相变的存在。     

Entanglement renormalization

We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block.     

Ellipses Percolation

We define a continuum percolation model that provides a collection of random ellipses on the plane and study the connectivity behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter \(\alpha > 0\) associated with the tail decay of the major axis distribution; we only consider distributions \(\rho \) satisfying \(\rho [r, \infty ) \asymp r^{-\alpha }\). We prove that this model presents a double phase transition in \(\alpha \). For \(\alpha \in (0,1]\) the plane is completely covered by the ellipses, almost surely. For \(\alpha \in (1,2)\) the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For \(\alpha \in (2, \infty )\) the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter \(\alpha = 2\) that there is a non-degenerate interval of densities for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on \(\mathbb {Z}^2\).     

Deterministic Percolation

This paper examines percolation questions in a deterministic setting. In particular, I consider , the set of elements of Z ² with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asymptotic density. An “almost everywhere” sieve of J. Friedlander is used to obtain the result. Received: 1 November 1998 / Accepted: 1 April 1999     

The equations of Wilson's renormalization group and analytic renormalization

In the present series of two papers we solve exactly Wilson's equations for a long-range effective hamiltonian. These equations arise when one seeks a fixed point of the Wilson's renormalization group transformations in the formulation of perturbation theory. The first paper has a general character. Wilson's renormalization transformation and its modifications are defined and the group property for them is established. Some topological aspects of the renormalization transformations are discussed. A space of projection hamiltonians is introduced and a theorem on the invariance of this space with respect to the renormalization transformations is proved.     

Electrostatic potential screened by a two-dimensional electron system: a real-space observation by scanning-tunneling spectroscopy

Scanning-tunneling spectroscopy at 5 K was used to investigate the electrostatic potential profile on the Si(111)-square root of 3 x square root of 3 Ag surface at subnanometer spatial resolution. The potential was measured from an energy-level shift of electronic states on the surface. The potential images obtained reveal that the potential drops around the steps and Ag adsorbates, upon which positive charges are presumably accumulated. The profiles of the reduced potentials are explained with the screening of potential due to the charges by two-dimensional electron gas (2DEG) existing on the surface. The Friedel oscillation, which results from the screening and has a period of the half Fermi wavelength of the 2DEG, was also observed in the potential images.