Backbends in Directed Percolation

When directed percolation in a bond percolation process does not occur, any path to infinity on the open bonds will zigzag back and forth through the lattice. Backbends are the portions of the zigzags that go against the percolation direction. They are important in the physical problem of particle transport in random media in the presence of a field, as they act to limit particle flow through the medium. The critical probability for percolation along directed paths with backbends no longer than a given length n is defined as p _n. We prove that (p _n) is strictly decreasing and converges to the critical probability for undirected percolation p _c. We also investigate some variants of the basic model, such as by replacing the standard d-dimensional cubic lattice with a (d–1)-dimensional slab or with a Bethe lattice; and we discuss the mathematical consequences of alternative ways to formalize the physical concepts of percolation and backbend.     

Tricritical Directed Percolation

We consider a modification of the contact process incorporating higher-order reaction terms. The original contact process exhibits a non-equilibrium phase transition belonging to the universality class of directed percolation. The incorporated higher-order reaction terms lead to a non-trivial phase diagram. In particular, a line of continuous phase transitions is separated by a tricritical point from a line of discontinuous phase transitions. The corresponding tricritical scaling behavior is analyzed in detail, i.e., we determine the critical exponents, various universal scaling functions as well as universal amplitude combinations. PACS numbers: 05.70.Ln, 05.50.+q, 05.65.+b     

Renormalized Field Theory of Resistor Diode Percolation

We study resistor diode percolation at the transition from the non-percolating to the directed percolating phase. We derive a field theoretic Hamiltonian which describes not only geometric aspects of directed percolation clusters but also their electric transport properties. By employing renormalization group methods we determine the average two-port resistance of critical clusters, which is governed by a resistance exponent . We calculate to two-loop order.     

Adsorbing and Collapsing Directed Animals

A model of a self-interacting directed animal, which also interacts with a solid wall, is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition. The directed animal is confined to a 45° wedge, and it interacts with one of the walls of this wedge. The existence of a thermodynamic limit in this model shown, and the presence of an adsorption transition is demonstrated by using constructive techniques. By comparing this model to a process of directed percolation, we show that there is also a collapse or -transition in this model. We examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity. The generating function of adsorbing directed animals in a half-space is found next from which we find the tricritical exponents associated with the adsorption transition. A full solution for a collapsing directed animal seems intractible, so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next.     

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.     

Continuum Percolation with Unreliable and Spread-Out Connections

We derive percolation results in the continuum plane that lead to what appears to be a general tendency of many stochastic network models. Namely, when the selection mechanism according to which nodes are connected to each other, is sufficiently spread out, then a lower density of nodes, or on average fewer connections per node, are sufficient to obtain an unbounded connected component. We look at two different transformations that spread-out connections and decrease the critical percolation density while preserving the average node degree. Our results indicate that real networks can exploit the presence of spread-out and unreliable connections to achieve connectivity more easily, provided they can maintain the average number of functioningconnections per node.     

First-Passage Percolation, Semi-Directed Bernoulli Percolation, and Failure in Brittle Materials

We present a two-dimensional, quasistatic model of fracture in disordered brittle materials that contains elements of first-passage percolation, i.e., we use a minimum-energy-consumption criterion for the fracture path. The first-passage model is employed in conjunction with a semi-directed Bernoulli percolation model, for which we calculate critical properties such as the correlation length exponent v ^sdir and the percolation threshold p _c ^sdir . Among other results, our numerics suggest that v ^sdir is exactly 3/2, which lies between the corresponding known values in the literature for usual and directed Bernoulli percolation. We also find that the well-known scaling relation between the wandering and energy fluctuation exponents breaks down in the vicinity of the threshold for semi-directed percolation. For a restricted class of materials, we study the dependence of the fracture energy (toughness) on the width of the distribution of the specific fracture energy and find that it is quadratic in the width for small widths for two different random fields, suggesting that this dependence may be universal.     

Partial dimensional sequences and percolation

We discuss a new mathematical approach of describing fractal lattices by means of transfer matrices of fractals (TMFs). These matrices have interesting mathematical properties. Possible physical applications of the TMFs are briefly indicated.     

Field Theory of Critical Behavior in Driven Diffusive Systems with Quenched Disorder

We present a field-theoretic renormalization-group study for the critical behavior of a uniformly driven diffusive system with quenched disorder, which is modeled by different kinds of potential barriers between sites. Due to their symmetry properties, these different realizations of the random potential barriers lead to three different models for the phase transition to transverse order and to one model for the phase transition to longitudinal order all belonging to distinct universality classes. In these four models, which have different upper critical dimensions d _c, we find the critical scaling behavior of the vertex functions in spatial dimensions d<d _c. The deviation from purely diffusive behavior is characterized by the anomaly exponent , which we calculate at first and second order, respectively, in =d _c–d. In each model turns out to be positive, which means superdiffusive spread of density fluctuations in the driving force direction.     

提高重整化群精度的一个尝试

以热逾渗分析散粒体导热率时,利用重整化群方法改变粗视化程度来定量地获得导热率的变化。实践表明这只有设法提高重整化群的精度才会有较好的结果。本文以逾渗转变为例,针对b=2的相关尺度变化方式,分别对二维和三维实空间的重整化变换进行了修正,导出了相应的重整化方程。计算精度有明显提高,计算结果与实验值更接近。     

Percolation,fractals, and anomalous diffusion

Both the infinite cluster and its backbone are self-similar at the percolation threshold,p _c . This self-similarity also holds at concentrationsp nearp _c , for length scalesL which are smaller than the percolation connectedness length,. ForL<, the number of bonds on the infinite cluster scales asL ^D , where the fractal dimensionalityD is equal to(d-/v). Geometrical fractal models, which imitate the backbone and on which physical models are exactly solvable, are presented. Above six dimensions, one has D=4 and an additional scaling length must be included. The effects of the geometrical structure of the backbone on magnetic spin correlations and on diffusion at percolation are also discussed.     

Self-avoiding walks in quenched random environments

The self-avoiding walk in a quenched random environment is studied using real-space and field-theoretic renormalization and Flory arguments. These methods indicate that the system is described, ford _c =4, and, for large disorder ford>d _c , by a strong disorder fixed point corresponding to a glass state in which the polymer is confined to the lowest energy path. This fixed point is characterized by scaling laws for the size of the walk,LN ^p withN the number of steps, and the fluctuations in the free energy,fL ^p. The bound 1/-d/2 is obtained. Exact results on hierarchical lattices yield> _pure and suggests that this inequality holds ford=2 and 3, although= _pure cannot be excluded, particularly ford=2. Ford>d _c there is a transition between strong and weak disorder phases at which= _pure. The strong-disorder fixed point for SAWs on percolation clusters is discussed. The analogy with directed walks is emphasized.     

Percolation on networks with dependence links

As a classical model of statistical physics, the percolation theory provides a powerful approach to analyze the network structure and dynamics. Recently, to model the relations among interacting agents beyond the connection of the networked system, the concept of dependence link is proposed to represent the dependence relationship of agents. These studies suggest that the percolation properties of these networks differ greatly from those of the ordinary networks. In particular,unlike the well known continuous transition on the ordinary networks, the percolation transitions on these networks are discontinuous. Moreover, these networks are more fragile for a broader degree distribution, which is opposite to the famous results for the ordinary networks. In this article, we give a summary of the theoretical approaches to study the percolation process on networks with inter- and inner-dependence links, and review the recent advances in this field, focusing on the topology and robustness of such networks.     

The Renormalization Group and Optimization of Entropy

We illustrate the possible connection that exists between the extremal properties of entropy expressions and the renormalization group (RG) approach when applied to systems with scaling symmetry. We consider three examples: (1) Gaussian fixed-point criticality in a fluid or in the capillary-wave model of an interface; (2) Lévy-like random walks with self-similar cluster formation; and (3) long-ranged bond percolation. In all cases we find a decreasing entropy function that becomes minimum under an appropriate constraint at the fixed point. We use an equivalence between random-walk distributions and order-parameter pair correlations in a simple fluid or magnet to study how the dimensional anomaly at criticality relates to walks with long-tailed distributions.     

On Percolation with Fibers or Layers

We consider site percolation on Z ^d, directed edges going from any sZ ^d to s+A ₁,..., s+A _n, where A ₁,..., A _n are the same for all sites and at least two of them are noncollinear. A site is closed if it belongs to p+Block, where p is a point in a Poisson distribution in R ^dZ ^d with a density and Block={sL: |s|M}+{sR ^d: |s|}, where L is a linear subspace of R ^d, |·| is the Euclidean norm, =max(|A ₁|,..., |A _n|) and M is a parameter. We study the behavior of *, the critical value, and P _closed*, corresponding critical percentage of closed sites, when M. Denote R ^d/L the factor space. Call two nonzero vectors U, V codirected if U=kV, where k>0. Theorem. If there are A _i and A _j whose projections to R ^d/L are not codirected, then *1/M ^dim(L) and P _closed* remains separated both from 0 and 1 when M. If projections of all A ₁,..., A _n to R ^d/L are codirected, then *1/M ^dim(L)+1 and P _closed*1/M when M.     

Magnetic correlations on fractals

The critical behavior of magnetic spin models on various fractal structures is reviewed, with emphasis on branching and nonbranching Koch curves and Sierpiriski gaskets and carpets. The spin correlation function is shown to have unusual exponential decays, e.g., of the form exp[-(r/gx) ^x ], and to crossover to other forms at larger distancesr. The various fractals are related to existing models for the backbone of the infinite incipient cluster at the percolation threshold, and conclusions are drawn regarding the behavior of spin correlations on these models.     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

化整为零:聚合物/粘土纳米复合材料的微观结构和阻隔性能

在聚合物基体中掺入少量的层状硅酸盐所制备的聚合物／粘土纳米复合材料，其阻隔性能明显地优于纯聚合物及其传统的复合材料。实验及分析结果表明，聚合物／粘土纳米复合材料的微观结构和阻隔性能主要受控于粘土剥离后的径厚比．一简单的重整化群模型被用来评估粘土几何因素（诸如径厚比、取向、剥离程度等）对聚合物／粘土纳米复合材料阻隔性能的影响，所得到的逾渗阈值及最佳粘土含量与实验结果吻合。     

Role of scaling in the statistical modelling of finance

Modelling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an autoregressive strategy. Results are fully consistent with volatility clustering and with the multiscaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.      

Real space renormalization group theory of the percolation model

A cluster expansion renormalization group method in real space is-developed to determine the critical properties of the percolation model. In contrast to previous renormalization group approaches, this method considers the cluster size distribution (free energy) rather than the site or bond probability distribution (coupling constants) and satisfies the basic renormalization group requirement of free energy conservation. In the construction of the renormalization group transformation, new couplings are generated which alter the topological structure of the clusters and which must be introduced in the original system. Predicted values of the critical exponents appear to converge to presumed exact values as higher orders in the expansion are considered. The method can in principle be extended to different lattice structures, as well as to different dimensions of space.This paper is dedicated to Prof. Philippe Choquard.