Long-range epidemic spreading in a random environment

Modeling long-range epidemic spreading in a random environment, we consider a quenched disordered, $d$-dimensional contact process with infection rates decaying with the distance as $1/r^{d+\sigma}$. We study the dynamical behavior of the model at and below the epidemic threshold by a variant of the strong-disorder renormalization group method and by Monte Carlo simulations in one and two spatial dimensions. Starting from a single infected site, the average survival probability is found to decay as $P(t) \sim t^{-d/z}$ up to multiplicative logarithmic corrections. Below the epidemic threshold, a Griffiths phase emerges, where the dynamical exponent $z$ varies continuously with the control parameter and tends to $z_c=d+\sigma$ as the threshold is approached. At the threshold, the spatial extension of the infected cluster (in surviving trials) is found to grow as $R(t) \sim t^{1/z_c}$ with a multiplicative logarithmic correction, and the average number of infected sites in surviving trials is found to increase as $N_s(t) \sim (\ln t)^{\chi}$ with $\chi=2$ in one dimension.Comment: 12 pages, 6 figure

Random transverse-field Ising chain with long-range interactions

We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power alpha of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent z_c=alpha. Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an alpha-independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.Comment: 6 pages, 4 figure

A new method for hardness determination from depth sensing indentation tests

A new semiempirical formula is developed for the hardness determination of the materials from depth sensing indentation tests. The indentation works measured both during loading and unloading periods are used in the evaluation. The values of the Meyer hardness calculated in this way agree well with those obtained by conventional optical observation, where this latter is possible. While the new hardness formula characterizes well the behavior of the conventional hardness number even for the ideally elastic material, the mean contact pressure generally used in hardness determination differs significantly from the conventional hardness number when the ideally elastic limiting case is being approached

Entanglement between random and clean quantum spin chains

The entanglement entropy in clean, as well as in random quantum spin chains has a logarithmic size-dependence at the critical point. Here, we study the entanglement of composite systems that consist of a clean and a random part, both being critical. In the composite, antiferromagnetic XX-chain with a sharp interface, the entropy is found to grow in a double-logarithmic fashion ${\cal S}\sim \ln\ln(L)$, where $L$ is the length of the chain. We have also considered an extended defect at the interface, where the disorder penetrates into the homogeneous region in such a way that the strength of disorder decays with the distance $l$ from the contact point as $\sim l^{-\kappa}$. For $\kappa<1/2$, the entropy scales as ${\cal S}(\kappa) \simeq (1-2\kappa){\cal S}(\kappa=0)$, while for $\kappa \ge 1/2$, when the extended interface defect is an irrelevant perturbation, we recover the double-logarithmic scaling. These results are explained through strong-disorder RG arguments.Comment: 12 pages, 7 figures, Invited contribution to the Festschrift of John Cardy's 70th birthda

Griffiths phases in the contact process on complex networks

Dynamical processes occurring on top of complex networks have become an exciting area of research. Quenched disorder plays a relevant role in general dynamical processes and phase transitions, but the effect of topological quenched disorder on the dynamics of complex networks has not been systematically studied so far. Here, we provide heuristic and numerical analyses of the contact process defined on some complex networks with topological disorder. We report on Griffiths phases and other rare region effects, leading rather generically to anomalously slow relaxation in generalized small-world networks. In particular, it is illustrated that Griffiths phases can emerge as the consequence of pure topological heterogeneity if the topological dimension of the network is finite.Comment: 5 pages, 2 figures, proc. of 11th Granada Seminar on Computational Physic

The effect of asymmetric disorder on the diffusion in arbitrary networks

Considering diffusion in the presence of asymmetric disorder, an exact relationship between the strength of weak disorder and the electric resistance of the corresponding resistor network is revealed, which is valid in arbitrary networks. This implies that the dynamics are stable against weak asymmetric disorder if the resistance exponent $\zeta$ of the network is negative. In the case of $\zeta>0$, numerical analyses of the mean first-passage time $\tau$ on various fractal lattices show that the logarithmic scaling of $\tau$ with the distance $l$, $\ln\tau\sim l^{\psi}$, is a general rule, characterized by a new dynamical exponent $\psi$ of the underlying lattice.Comment: 5 pages, 4 figure

Anomalous diffusion in disordered multi-channel systems

We study diffusion of a particle in a system composed of K parallel channels, where the transition rates within the channels are quenched random variables whereas the inter-channel transition rate v is homogeneous. A variant of the strong disorder renormalization group method and Monte Carlo simulations are used. Generally, we observe anomalous diffusion, where the average distance travelled by the particle, []_{av}, has a power-law time-dependence []_{av} ~ t^{\mu_K(v)}, with a diffusion exponent 0 \le \mu_K(v) \le 1. In the presence of left-right symmetry of the distribution of random rates, the recurrent point of the multi-channel system is independent of K, and the diffusion exponent is found to increase with K and decrease with v. In the absence of this symmetry, the recurrent point may be shifted with K and the current can be reversed by varying the lane change rate v.Comment: 16 pages, 7 figure

Slow dynamics of the contact process on complex networks

The Contact Process has been studied on complex networks exhibiting different kinds of quenched disorder. Numerical evidence is found for Griffiths phases and other rare region effects, in Erd˝os Rényi networks, leading rather generically to anomalously slow (algebraic, logarithmic,...) relaxation. More surprisingly, it turns out that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of sole topological heterogeneity in networks with finite topological dimension. In case of scalefree networks, exhibiting infinite topological dimension, slow dynamics can be observed on tree-like structures and a superimposed weight pattern. In the infinite size limit the correlated subspaces of vertices seem to cause a smeared phase transition. These results have a broad spectrum of implications for propagation phenomena and other dynamical process on networks and are relevant for the analysis of both models and empirical data