Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

Conductance distributions in random resistor networks. Self-averaging and disorder lengths

The self-averaging properties of the conductanceg are explored in random resistor networks (RRN) with a broad distribution of bond strengthsP(g)g ^–1. The RRN problem is cast in terms of simple combinations of random variables on hierarchical lattices. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of sizeL and the distribution tail strength parameter . For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit 0. Adisorder length _D is identified, beyond which the system is effectively homogeneous. This length scale diverges as _Dµ^–v ( is the regular percolation correlation length exponent) when the microscopic distribution of conductors is exponentially wide (0). This implies that exactly the same critical behavior can be induced by geometrical disorder and by strong bond disorder with the bond occupation probabilityp. We find that only lattices at the percolation threshold have renormalized probability distributions in aLevy-like basin. At the percolation threshold the disorder length diverges at a critical tail strength µ_c as µ–^–z withz3.2±0.1, a new exponent.Critical path analysis is used in a generalized form to give the macroscopic conductance in the case of lattices abovep _c.     

Scaling law of resistance fluctuations in stationary random resistor networks

In a random resistor network we consider the simultaneous evolution of two competing random processes consisting in breaking and recovering the elementary resistors with probabilities W(D) and W(R). The condition W(R)>W(D)/(1+W(D)) leads to a stationary state, while in the opposite case, the broken resistor fraction reaches the percolation threshold p(c). We study the resistance noise of this system under stationary conditions by Monte Carlo simulations. The variance of resistance fluctuations is found to follow a scaling law |p-p(c)|(-kappa(0)) with kappa(0) = 5.5. The proposed model relates quantitatively the defectiveness of a disordered media with its electrical and excess-noise characteristics.     

Excess thermal-noise in the electrical breakdown of random resistor networks

We discuss a new type of excess noise strongly sensitive to non-homogeneous Joule heating of random resistor network and associated with local sources of thermal noise. The evolution of the network towards an electrical breakdown of conductor-insulator type is then studied by using a biased percolation model and it is analysed in terms of an excess-noise temperature. Monte Carlo simulation results show a significant increase of the excess-noise temperature over the average temperature of the network. Remarkably the excess-noise temperature scales with the resistance with an exponent of about 3. The predictivity of the model can be tested on thin film resistors where the determination of the excess noise temperature should provide a valuable indicator of the defectiveness of the film. Received 13 April 1999 and Received in final form 7 May 1999     

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP (p) is discontinuous atp _c , then the critical exponent (defined by the divergence of expected cluster size, nP _n (p) (P _c –P)^– asp p _c ) must satisfy 2. (2) or (defined analogously to, but asp p _c ) and [P _n (p _c ) (n ^–1–1/) asn ] must satisfy, 2(1 – 1/). These inequalities for improve the previously known bound 1(Aizenman and Newman), since 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P _x has no discontinuity atp _c .     

Stability and dynamical properties of material flow systems on random networks

The theory of complex networks and of disordered systems is used to study the stability and dynamical properties of a simple model of material flow networks defined on random graphs. In particular we address instabilities that are characteristic of flow networks in economic, ecological and biological systems. Based on results from random matrix theory, we work out the phase diagram of such systems defined on extensively connected random graphs, and study in detail how the choice of control policies and the network structure affects stability. We also present results for more complex topologies of the underlying graph, focussing on finitely connected Erdös-Réyni graphs, Small-World Networks and Barabási-Albert scale-free networks. Results indicate that variability of input-output matrix elements, and random structures of the underlying graph tend to make the system less stable, while fast price dynamics or strong responsiveness to stock accumulation promote stability.     

A transfer matrix program to calculate the conductivity of random resistor networks

We describe technical details on a new method of calculating the conductivity of random resistor networks which uses transfer matrix ideas. We give a program which calculates the conductivity of three-dimensional bars, and we provide a few comments on the advantages of this method and its performances.     

A lower bound on the variance of conductance in random resistor networks

We study the conductance of random resistor networks ind2 dimensions. It is shown (under some technical assumptions) that if a network exhibits a nonzero conductivity in the infinite-volume limit, then the variance of a finite-volume conductance grows at least like the volume.     

A finite-time exponent for random Ehrenfest gas

We consider the motion of a system of free particles moving on a plane with regular hard polygonal scatterers arranged in a random manner. Calling this the Ehrenfest gas, which is known to have a zero Lyapunov exponent, we propose a finite-time exponent to characterize its dynamics. As the number of sides of the polygon goes to infinity, when polygon tends to a circle, we recover the usual Lyapunov exponent for the Lorentz gas from the exponent proposed here. To obtain this result, we generalize the reflection law of a beam of rays incident on a polygonal scatterer in a way that the formula for the circular scatterer is recovered in the limit of infinite number of vertices. Thus, chaos emerges from pseudochaos in an appropriate limit.     

Single-site approximations for equilibrium and transport properties of random alloys

Introducing the configuration averaging as a projection operator, we propose a new formulation of the average t-matrix and the coherent potential approximations that treats on equal footing both the equilibrium and the residual transport properties of disordered alloys.     

La0.7Ca0.3 MnO3和CeO2混合块状样品电输运性质及使用分形迭代电阻网络模型的计算模拟

固相反应方法制得的La0.7Ca0.3MnO3(LCMO)粉末与CeO2粉末均匀混合后压块烧结,对其进行了电阻率随温度变化的测量.观察到样品电阻率存在绝缘体-金属转变.研究了烧结工艺、CeO2的混合比例与样品的电阻率、结构和表面形貌的关系.结果表明,制备工艺对材料性能有相当大的影响,其中较低温度的烧结主要影响小晶粒及其晶粒间界,而高温长时间烧结将影响大晶粒的形成.利用三维随机电阻网络(RRN)模型和蒙特卡罗方法对这种混合块状样品的输运性质进行了模拟,模拟中使用了一种新的RRN平均方法.该模型得到的数值模拟结果与实验结果在定性上是一致的.这说明以RRN模型来理解LCMO(x)-CeO2(1-x)复合体系的导电状况是合理的,提出的随机电阻网络平均方法是合适的.     

Theta-point exponent for polymer chain in random media

Using field-theoretic arguments for self-avoiding walks on dilute lattices with site occupation concentrationp, we show that the-point size exponent _p ⁰ of polymer chains remains unchanged for small disorder concentration (p>p _c ). At the percolation thresholdp=p _c , using a Flory-type approximation, we conjecture that _pc ⁰ =5/(d _B +7), whered _B is the percolation backbone dimension. It shows that the upper critical dimensionality for the-point transition atp=p _c shifts to a dimensiond _c >3. We also propose that the-point varies practically linearly withp for 1>pp _c .     

On the critical exponent for random walk intersections

The exponent _d for the probability of nonintersection of two random walks starting at the same point is considered. It is proved that 1/2<₂3/4. Monte Carlo simulations are done to suggest ₂=0.61 and ₃0.29.