Quenched Invariance Principles for Random Walks with Random Conductances

We prove an almost sure invariance principle for a random walker among i.i.d. conductances in ℤ ^d , d≥2. We assume conductances are bounded from above but we do not require that they are bounded from below.     

Random Phases and Energy Dispersion

Using 2N 1 successive stationary states centred at nth,we construct a rectangular wavepacket in which the stationary states are superimposed with the equal weight 2N 1.With the requirement of the wavepacket to be a quasi-classical state,the number N is determined by minimizing the uncertainty ΔxΔp,Since the stationary state can only be determined to within an arbitrary multiplicative complex phase factor of unit magnitude,a number of N is obtained as a set of the phases are given.For a harmonic oscillator,when all of the phase factors are essentially the same,we have N≈[6^1/3n^2/3] with [x] signifying the integral part of positive number x,When every phase in the phase factors is given by a random number generated in a closed interval [0,2π] and when n≥10,the probability of appearance of N is roughly 1/2^N when N=1 to 7,and does not exceed 0.01 when N≥8.     

Recurrence and Transience of Random Walks¶in Random Environments on a Strip

We explain the necessary and sufficient conditions for recurrent and transient behavior of a random walk in a stationary ergodic random environment on a strip in terms of properties of a top Lyapunov exponent. This Lyapunov exponent is defined for a product of a stationary sequence of positive matrices. In the one-dimensional case this approach allows us to treat wider classes of random walks than before. Received: 15 March 2000 / Accepted: 14 April 2000     

Kardar–Parisi–Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

We consider the transition probabilities for random walks in \(1+1\) dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.     

Random‐field experiments in dilute antiferromagnets

Experiments in d = 2 and d = 3 random-field Ising systems are reviewed. Random fields destroy the phase transition for the d = 2 system Rb₂Co _x Mg_1-xF₄. The d = 3 Fe _x Zn_1-xF₄ system shows a new phase transition with unusual dynamics. Extreme critical slowing down plays an essential role in the d = 3 measurements. Experiments on the related d = 3 systems Mn _x Zn_1-xF₂ and Fe _x Mg_1-xC1₂ are also discussed.     

Random and Preferential Attachment Networks with Aging

A new model for network generation is proposed by considering both the preferential and random attachments with aging. The connectivity distribution is obtained with the mean-field theory. The simulation results show that the model can be used to generate such networks with different topology structures as random and scale-free ones. The networks with different densities (i.e., the average connectivity degrees) can also be generated by the model.     

White Noise in Quantum Random Walk Search Algorithm

The quantum random walk is a possible approach to construct new quantum search algorithms. It has been shown by Shenvi et al. [Phys. Rev. A 67（2003）52307] that a kind of algorithm can perform an oracle search on a database of N items with O（√N） calling to the oracle, yielding a speedup similar to other quantum search algorithms. We study the effect of white or Gaussian noise on this algorithm. The algorithm loses efficiency when noise is added. We also show that noise on the target state plays a more important role than that on other states. Finally we compare the effects of similar types of noise in the quantum random walk search algorithm and Grover＇s search algorithm.     

Avalanche Dynamics in Quenched Random Directed Sandpile Models

We numerically investigate the quenched random directed sandpile models which are local, conservative and Abelian. A local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site plays an important role when all the nearest-neighbouring sites of the above-mentioned site topple for once. The quenched model has the same critical exponents with the Abelian deterministic directed sandpile model when the local flow balance exists, otherwise the critical exponents of this quenched model and the annealed Abelian random directed sandpile model are the same. These results indicate that the presence or absence of this local flow balance determines the universality class of the Abelian directed sandpile model.     

Random response of an oscillator with hysteresis damping

The response behaviour of an oscillator with Reid hysteresis damping un-der stationary Gauss white excitation is studied.By means of numerical simulationconducted on computer,the mean square response,probability density function andpower spectral density function for the response are given.These response character-istics are compared with those predicted by equivalent linearization treatment.     

Extended Quark Potential Model From Random Phase Approximation

The quark potential model is extended to include the sea quark excitation using the random phase approximation.The effective quark interaction preserves the important QCD properties-chiral symmetry and confinement simultaneously.A primary qualitative analysis shows that the π meson as a well-known typical Goldstone boson and the other mesons made up of valence qq quark pair such as the ρ meson can also be described in this extended quark potential model.     

Characteristic Polynomials¶of Real Symmetric Random Matrices

The correlation functions of the random variables det(λ−X), in which X is an hermitian N×N random matrix, are known to exhibit universal local statistics in the large N limit. We study here the correlation of those same random variables for real symmetric matrices (GOE). The derivation relies on an exact dual representation of the problem: the k-point functions are expressed in terms of finite integrals over (quaternionic) k×k matrices. However the control of the Dyson limit, in which the distance of the various parameters λ's is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson–Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a finite number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large N limit. Received: 19 March 2001 / Accepted: 21 June 2001     

Random fiber laser operating at 1,115 nm

We experimentally demonstrate a random fiber laser operating at 1,115 nm using a LD-pumped Yb-doped fiber laser as the pump source. We achieve about 270 mW lasing output in a 50 km standard communication optical fiber with slope efficiency more than 28 %. A new wavelength is provided for the application of random distributed feedback fiber lasers as light sources.     

Mode-Locking Behaviour in Driven Colloids with Random Pinning

We find mode-locking steps in simulated force-velocity characteristics of external alternating-force （AF） driven colloids on a disordered substrate. Studies of mode-locking patterns in systems show that mode-locking steps are accompanied with the emergence of a dynamics phase： transverse solid phase. We also study the influence of temperature on the width of mode-locking steps. The mode-locked state is destroyed by thermal fluctuation and the width of mode-locking steps decreases rapidly with increasing temperature. In high velocity and low temperature regimes, due to the appearance of transverse solid phase and microscopically periodic velocity modulation, the step width changes little as temperature is varied.     

Random Unitary Matrices, Permutations and Painlevé

This paper is concerned with certain connections between the ensemble of n×n unitary matrices – specifically the characteristic function of the random variable tr(U) – and combinatorics – specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups – and the appearance of Painlevé functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups. Received: 2 December 1998 / Accepted: 12 May 1999     

Soliton Propagation in Optical Fibers with Random Polarization Dispersion

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Multistage Random Growing Small-World Networks with Power-Law Degree Distribution

We present a simple rule which could generate scale-free networks with very large clustering coefficient and very small average distance. These networks, called the multistage random growing networks （MRGNs）, are constructed by a two-stage adding process for each new node. The analytic results of the power-law exponent = 3 and the clustering coefficient C = 0.81 are obtained, which agree with the simulation results approximately. In addition, we find that the average distance of the networks increases logarithmically with the network size, which is consistent with the theoretical predictions. Since many real-world networks are both scale-free and small-world, the MRGNs may perform well in mimicking reality.     

Random walk in random environment: A counterexample?

We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimensiond<.This author partially supported by NSF grant DMS 83-1080This author partially supported by NSF grant DMS-85-05020 and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e ^{− L ζ}, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions. Received: 29 November 2000 / Accepted: 21 June 2001     

Matérn Class Tensor-Valued Random Fields and Beyond

We construct classes of homogeneous random fields on a three-dimensional Euclidean space that take values in linear spaces of tensors of a fixed rank and are isotropic with respect to a fixed orthogonal representation of the group of \(3\times 3\) orthogonal matrices. The constructed classes depend on finitely many isotropic spectral densities. We say that such a field belongs to either the Matérn or the dual Matérn class if all of the above densities are Matérn or dual Matérn. Several examples are considered.