Markov chains are an important idea, related to random walks, which crops up widely in applied stochastic analysis. They are used, for example, in performance modelling and evaluation of computer networks, queuing networks, and telecommunication systems. The main point of the present book is to provide methods, based on the construction of Lyapunov functions, of determining when a Markov chain is ergodic, null recurrent, or transient. These methods can also be extended to the study of questions of stability. Of particular concern are reflected random walks and reflected Brownian motion. The authors provide not only a self-contained introduction to the theory but also details of how the required Lyapunov functions are constructed in various situations.

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The single-parameter scaling hypothesis predicts the absence of delocalized states for noninteracting quasiparticles in low-dimensional disordered systems. We show analytically, using a supersymmetric method combined with a renormalization group analysis, as well as numerically that extended states may occur in the one-and two-dimensional Anderson model with a nonrandom hopping falling off as some power of the distance between sites. The different size scaling of the bare level spacing and the renormalized magnitude of the disorder seen by the quasiparticles finally results in the delocalization of states at one of the band edges of the quasiparticle energy spectrum. DOI: 10.1103/PhysRevLett.90.027404 PACS numbers: 78.30.Ly, 36.20.Kd, 71.30.+h, 71.35.Aa Localization of noninteracting quasiparticles in random media with time-reversal symmetry and finiterange hopping have been extensively studied since the seminal paper by Anderson [1]. The hypothesis of singleparameter scaling, introduced in Ref.[2], led to the general belief that all eigenstates of noninteracting quasiparticles were exponentially localized in one (1D) and two (2D) dimensions (see Refs. [3,4] for a comprehensive review) and that localization-delocalization transitions no longer exist in the thermodynamic limit. Even though models with finite-range hopping work nicely in describing a variety of materials, long-range hopping is often found in different physical systems (e.g., Frenkel excitons). Random long-range hopping was found to give rise to delocalization of states not only in threedimensional systems [1] but also in any dimension [5][6][7][8]. Recent studies [9] revised the validity of the singleparameter scaling hypothesis even within the original 1D Anderson model with nearest-neighbor coupling, although did not question the statement that all eigenstates in 1D random systems are localized.In this Letter, we present analytical and numerical proofs that a localization-delocalization transition may occur in 1D and 2D systems with diagonal disorder and nonrandom intersite coupling which falls off according to a powerlike law. Apart from the importance of this finding from a general point of view, it may be relevant for several physical systems. As an example, let us mention dipolar Frenkel excitons on 2D regular lattices where molecules are subjected to randomness due to a disordered environment [10]. Biological light-harvesting antenna systems represent a realization of the model we are dealing with [11,12]. Magnons in 1D and 2D disordered spin systems provide one more example of interest.We consider the Anderson Hamiltonian on a d-dimensional (d 1;2) simple lattice with N N d sites:where jni is the ket-vector of the state localized at site n, and f" n g are random site energies, assumed to be uncorrelated for different sites and distributed uniformly within an interval ÿ=2;=2, thus having zero mean and standard deviation = 12 p . The hopping integrals between lattice sites m and n will be taken in the form J mn J=jm ÿ nj J mm 0, where J ...