We present a full analytical solution for the localisation length in the one-dimensional Anderson model with weak diagonal disorder in the vicinity of the band centre. The results are obtained with the Hamiltonian map approach that turns out to be more effective than other known methods. The analytical expressions are supported by numerical data. We also discuss the implications of our results for the single-parameter scaling hypothesis.Although it was introduced more than fifty years ago [1], the tight-binding model named after Anderson is still widely studied because it combines an elementary mathematical structure with non-trivial physical features. The simplicity of the definition notwithstanding, a complete analytical understanding of the model is quite difficult to obtain, even in the one-dimensional (1D) case which is the most amenable to analytical treatment. Some fundamental properties of the 1D Anderson model, however, have long been known. In particular, it was rigorously proved that all eigenstates are localised in the 1D Anderson model [2], unless the random potential exhibits specific spatial correlations (see [3] for a comprehensive treatment of localisation in models with correlated disorder).The localisation length is the key physical parameter which determines the spatial extension of the electronic states. A general formula for the localisation length in the 1D Anderson model is still not known, but expressions for the limit cases of strong and weak disorder have been derived. The weakdisorder case, in particular, can be studied using a perturbative approach due to Thouless [4]. Thouless' method gives a formula for the inverse localisation length which works very well for most energies inside the band of the disorder-free model. This perturbative formula, however, has its flaws: in fact, shortly after the publication of Ref. [4], numerical calculations showed that Thouless' expression does not reproduce the correct value of the localisation length at the band centre, i.e., for E = 0 [5]. A short time later, Kappus and Wegner were able to ascribe this discrepancy to a resonance effect which leads to a breakdown of the standard perturbation theory [6]. Almost at the same time, and without any explicit connection to the work of Ref. [5,6], an analytical expression of the localisation length for E = 0 was derived in [7].A thorough study of the "anomaly" at the band centre was eventually performed by Derrida and Gardner [8]. These authors showed that a "naive" perturbative approach was bound to fail not only at the band centre, but also for every other "rational" value of the energy, i.e., for E = 2 cos(πr) with r a rational number. Such an approach, in fact, gives an expansion of the localisation length which breaks down because some coefficients diverge. To avoid this pitfall, Derrida and Gardner devised a specific perturbative technique which allowed them to analyse the anomalous behaviour of the localisation length in the neighbourhood of the band centre. In particular, Derrida

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1α(G) and M2α(G), and the general sum-connectivity index, χα(G)) as well as of general versions of indices of interest: the general inverse sum indeg index ISIα(G) and the general first geometric-arithmetic index GAα(G) (with α∈R). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks GER(nER,p) and random geometric (RG) graphs GRG(nRG,r). The ER random networks are formed by nER vertices connected independently with probability p∈[0,1]; while the RG graphs consist of nRG vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius r∈[0,2]. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree k of the corresponding random network models, where kER=(nER−1)p and kRG=(nRG−1)(πr2−8r3/3+r4/2). That is, X(GER)/nER≈X(GRG)/nRG if kER=kRG, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.

We analyse the thermal properties of a harmonic chain with weak correlated disorder. With the use of a perturbative approach we derive analytical expressions for the time-evolution of the chain temperature and of the heat flow when both ends of the chain are coupled to heat baths. Our analytical and numerical results demonstrate that specific long-range correlations of the isotopic disorder can suppress or enhance the vibrational modes in pre-defined frequency windows. In this way one can arrange a frequency-selective heat flow through disordered chains.

We address the general problem of heat conduction in one dimensional harmonic chain, with correlated isotopic disorder, attached at its ends to white noise or oscillator heat baths. When the low wavelength µ behavior of the power spectrum W (of the fluctuations of the random masses around their common mean value) scales as W (µ) ∼ µ β , the asymptotic thermal conductivity κ scales with the system size N as κ ∼ N (1+β)/(2+β) for free boundary conditions, whereas for fixed boundary conditions κ ∼ N (β−1)/(2+β) ; where β > −1, which is the usual power law scaling for one dimensional systems. Nevertheless, if W does not scale as a power law in the low wavelength limit, the thermal conductivity may not scale in its usual form κ ∼ N α , where the value of α depends on the particular one dimensional model. As an example of the latter statement, if W (µ) ∼ exp(−1/µ)/µ 2 , κ ∼ N/(log N ) 3 for fixed boundary conditions and κ ∼ N/ log(N ) for free boundary conditions, which represent non-standard scalings of the thermal conductivity.

We study the band-center anomaly in the one-dimensional Anderson model with the disorder characterized by short-range positive correlations. Using the Hamiltonian map approach, we obtain analytical expressions for the localization length and the invariant measure of the phase variable. The analytical expressions are complemented by numerical data.