Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight P_i ∝ i^-μ (0 < μ< 1) and an edge can connect verticesi andj with rateP _i P _j . Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, whereλ = 1 +μ ^-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.     

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs. It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type. For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1. We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases. Received: 27 March 2000 / Accepted: 7 December 2000     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

Critical Behaviors and Universality Classes of Percolation Phase Transitions on Two-Dimensional Square Lattice

We have investigated both site and bond percolation on two-dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.     

Inequalities in Entanglement Percolation

We present proofs of two results concerning entanglement in three-dimensional bond percolation. Firstly, the critical probability for entanglement with free boundary conditions is strictly less than the critical probability for connectivity percolation. (The proof presented here is a detailed justification of the ideas sketched in Aizenman and Grimmett.) Secondly, under the hypothesis that the critical probabilities for entanglement with free and wired boundary conditions are different, for p between the two critical probabilities, the size of the entangled cluster at the origin with free boundary conditions does not have exponentially decaying tails.     

On the uniqueness of the infinite cluster in the percolation model

We simplify the recent proof by Aizenman, Kesten and Newman of the uniqueness of the infinite open cluster in the percolation model. Our new proof is more suitable for generalization in the direction of percolation-type processes with dependent site variables.     

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 effects of current-path patterns on magnetotransport in spatially-confined structures by Monte Carlo simulation

Simulations are performed on clusters of finite size to study the effects of size and current-path structure on magnetotransport in spatially-confined samples. Magnetotransport networks are established and calculated based on fractal structures including Koch curves and percolation backbones extracted from regular lattices. The structure pattern of clusters is shown to play an important role in the magnetotransport behaviours by affecting the magnetoresistance fluctuations due to spin disorder in the systems of small size, which suggests the possibility of controlling the magnetotransport by the design of current-path configurations.     

Continuity of the Integrated Density of States on Random Length Metric Graphs

We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states. These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.      

含随机裂纹网络孔隙材料渗透率的逾渗模型研究

采用逾渗理论对含随机裂纹网络的孔隙材料渗透性进行研究. 开裂孔隙材料渗透率的影响因素包括裂纹网络的几何特征、孔隙材料本体渗透率以及裂纹开度, 本文使用连续区逾渗理论模型建立了渗透率的标度律. 对于裂纹网络的几何特征, 本文基于连续区逾渗理论并考虑裂纹网络的分形特征提出了有限区域内二维随机裂纹网络的连通度定义; 对随机裂纹网络的几何分析表明, 随机裂纹局部团簇效应会降低裂纹网络的整体连通性, 随机裂纹网络的标度指数并非经典逾渗理论给出的固定值, 而是随着网络的分形维数的减小而增大. 本文在网络连通度和主裂纹团的曲折度的基础上, 提出了开裂孔隙材料渗透率标度律的解析表达, K=K₀(K_m,b)(ρ-ρ_c)^μ, 分别考虑了裂纹网络的几何逾渗特征 (ρ-ρ_c)^μ、孔隙材料渗透率K_m 以及裂纹开度比b; 对有限区域含有随机裂纹网络的孔隙材料渗透过程的有限元模拟表明, K₀ 在裂纹逾渗阈值附近与b呈指数关系, 但当裂纹的局部渗透率与K_m比值高于10⁶ 后, 开度比b对渗透率不再有影响.     

Erdös Rényi随机网络上爆炸渗流模型相变性质的数值模拟研究

基于改进的Newman和Ziff算法以及有限尺寸标度理论, 通过对表征渗流相变特征物理量的序参量、平均集团尺寸、二阶矩、标准偏差及尺寸不均匀性的数值模拟, 分析研究了Erdös Rényi随机网络上Achlioptas爆炸渗流模型的相变性质.研究表明: 尽管序参量表现出了不连续相变的特征, 但序参量以及其他特征物理量仍具有连续相变的幂律标度行为.因此严格地说, Erdös Rényi随机网络中的爆炸渗流相变是一种奇异相变, 它既不是标准的不连续相变, 又与常规随机渗流表现出的连续相变处于不同的普适类. 关键词： Erdös Rényi随机网络 爆炸渗流模型 相变 幂律标度行为     

Analyticity and Mixing Properties for Random Cluster Model with q >0 on ℤ d

We study the Random Cluster Model on ℤ ^d for p near either 0 or 1 and for all q > 0 and we prove by mean of cluster expansion methods the analyticity of the pressure and finite connectivities in both regimes. These results are valid also in the regime q < 1 and they imply that percolation probability is strictly less than 1.     

A New Class of Cellular Automata with a Discontinuous Glass Transition

We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density ρ _c for convergence to a completely empty configuration is non trivial, 0<ρ _c <1, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, ρ<ρ _c , emptying always occurs exponentially fast and that ρ _c coincides with the critical density for two-dimensional oriented site percolation on ℤ². This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is discontinuous and at the same time the crossover length diverges faster than any power law. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques. Finally, we will discuss why, due to the peculiar mixed critical/first order character of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D.S. Fisher.     

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star–triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the q > 1 random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.     

A Simple Asymmetric Evolving Random Network

We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. We show that the model exhibits a percolation transition and discuss its universality. Below the threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent depending on the average connectivity. We prove that the transition is of infinite order by deriving the exact asymptotic formula for the size of the giant component close to the threshold. We also present a thorough analysis of aging properties. We compute local-in-time profiles for the components of finite size and for the giant component, showing in particular that the giant component is always dense among the oldest nodes but invades only an exponentially small fraction of the young nodes close to the threshold.     

Local Neighbourhoods for First-Passage Percolation on the Configuration Model

We consider first-passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.     

Growth, percolation, and correlations in disordered fiber networks

This paper studies growth, percolation, and correlations in disordered fiber networks. We start by introducing a 2D continuum deposition model with effective fiber-fiber interactions represented by a parameterp which controls the degree of clustering. Forp=1 the deposited network is uniformly random, while forp=0 only a single connected cluster can grow. Forp=0 we first derive the growth law for the average size of the cluster as well as a formula for its mass density profile. Forp>0 we carry out extensive simulations on fibers, and also needles and disks, to study the dependence of the percolation threshold onp. We also derive a mean-field theory for the threshold nearp=0 andp=1 and find good qualitative agreement with the simulations. The fiber networks produced by the model display nontrivial density correlations forp<1. We study these by deriving an approximate expression for the pair distribution function of the model that reduces to the exactly known case of a uniformly random network. We also show that the two-point mass density correlation function of the model has a nontrivial form, and discuss our results in view of recent experimental data on mss density correlations in paper sheets.     

Dynamical method for studying localization-delocalization transition in two dimensional percolating systems

We study quantum percolation which is described by a tight-binding Hamiltonian containing only off-diagonal hopping terms that are generally in quenched binary disorder (zero or one). In such a system, transmission of a quantum particle is determined by the disorder and interference effects, leading to interesting sharp features in conductance as the energy, disorder, and boundary conditions are varied. To aid understanding of this phenomenon, we develop a visualization method whereby the progression of a wave packet entering the cluster through a lead on one side and exiting from another lead on the other side can be tracked dynamically. Using this method, we investigate the localization-delocalization transition in a 2D system for various boundary conditions. Our results indicate the existence of two different kinds of localized regimes, namely exponential and power law localization, depending on the amount of disorder. Our study further suggests that there may be a delocalized state in the 2D quantum percolation system at very low disorder. These results are based on a finite size scaling analysis of the systems of size up to 70 × 70 (containing 4900 sites) on the square lattice.     

Percolation Phase Transitions from Second Order to First Order in Random Networks

We investigate a percolation process where an additional parameter q is used to interpolate between the classical Erd¨os–R′enyi(ER) network model and the smallest cluster(SC) model. This model becomes the ER network at q = 1, which is characterized by a robust second order phase transition. When q = 0, this model recovers to the SC model which exhibits a first order phase transition. To study how the percolation phase transition changes from second order to first order with the decrease of the value of q from 1 to 0, the numerical simulations study the final vanishing moment of the each existing cluster except the N-cluster in the percolation process. For the continuous phase transition,it is shown that the tail of the graph of the final vanishing moment has the characteristic of the convexity. While for the discontinuous phase transition, the graph of the final vanishing moment possesses the characteristic of the concavity.Just before the critical point, it is found that the ratio between the maximum of the sequential vanishing clusters sizes and the network size N can be used to decide the phase transition type. We show that when the ratio is larger than or equal to zero in the thermodynamic limit, the percolation phase transition is first or second order respectively. For our model, the numerical simulations indicate that there exists a tricritical point qcwhich is estimated to be between0.2 qc 0.25 separating the two phase transition types.     

Killing the Hofstadter butterfly,one bond at a time

Electronic bands in a square lattice when subjected to a perpendicular magnetic field form the Hofstadter butterfly pattern. We study the evolution of this pattern as a function of bond percolation disorder (removal or dilution of lattice bonds). With increasing concentration of the bonds removed, the butterfly pattern gets smoothly decimated. However, in this process of decimation, bands develop interesting characteristics and features. For example, in the high disorder limit, some butterfly-like pattern still persists even as most of the states are localized. We also analyze, in the low disorder limit, the effect of percolation on wavefunctions (using inverse participation ratios) and on band gaps in the spectrum. We explain and provide the reasons behind many of the key features in our results by analyzing small clusters and finite size rings. Furthermore, we study the effect of bond dilution on transverse conductivity (σ _xy ). We show that starting from the clean limit, increasing disorder reduces σ _xy to zero, even though the strength of percolation is smaller than the classical percolation threshold. This shows that the system undergoes a direct transition from a integer quantum Hall state to a localized Anderson insulator beyond a critical value of bond dilution. We further find that the energy bands close to the band edge are more stable to disorder than at the band center. To arrive at these results we use the coupling matrix approach to calculate Chern numbers for disordered systems. We point out the relevance of these results to signatures in magneto-oscillations.