Nature of largest cluster size distribution at the percolation threshold

Two distinct distribution functions $P_{sp}(m)$ and $P_{ns}(m)$ of the scaled largest cluster sizes $m$ are obtained at the percolation threshold by numerical simulations, depending on the condition whether the lattice is actually spanned or not. With $R(p_c)$ the spanning probability, the total distribution of the largest cluster is given by $P_{tot}(m) = R(p_c)P_{sp}(m) + (1-R(p_c))P_{ns}(m)$. The three distributions apparently have similar forms in three and four dimensions while in two dimensions, $P_{tot}(m)$ does not follow a familiar form. By studying the first and second cumulants of the distribution functions, the different behaviour of $P_{tot}(m)$ in different dimensions may be quantified.Comment: 7 pages revtex, figures included; to be published in J. Phys.

Quantum fluctuation induced spatial stochastic resonance at zero temperature

We consider a model in which the quantum fluctuation can be controlled and show that the system responds to a spatially periodic external field at zero temperature. This signifies the occurrence of spatial stochastic resonance where the fluctuations are purely quantum in nature. Various features of the phenomenon are discussed.Comment: To appear in Physical Review

Complexities of social networks: A Physicist's perspective

The review is a survey of the present status of research in social networks highlighting the topics of small world property, degree distributions, community structure, assortativity, modelling, dynamics and searching in social networks.Comment: 17 pages, an updated version will be published as a chapter in the book Econophysics and Sociophysics: Trends and perspectives, ed. B. K. Chakrabarti, A. Chakrabarti and A. Chatterjee, Wiley-VCH 200

Unusual scaling in a discrete quantum walk with random long range steps

A discrete time quantum walker is considered in one dimension, where at each step, the translation can be more than one unit length chosen randomly. In the simplest case, the probability that the distance travelled is $\ell$ is taken as $P(\ell) = \alpha \delta(\ell-1) + (1-\alpha) \delta (\ell-2^n)$ with $n \geq 1$. Even the $n=1$ case shows a drastic change in the scaling behaviour for any $\alpha \neq 0,1$. Specifically, $\langle x^2\rangle \propto t^{3/2}$ for $0 < \alpha < 1$, implying the walk is slower compared to the usual quantum walk. This scaling behaviour, which is neither conventional quantum nor classical, can be justified using a simple form for the probability density. The decoherence effect is characterized by two parameters which vanish in a power law manner close to $\alpha =0$ and $1$ with an exponent $\approx 0.5$. It is also shown that randomness is the essential ingredient for the decoherence effect.Comment: 15 pages, 10 figures, version accepted in Physica

Application of the Interface Approach in Quantum Ising Models

We investigate phase transitions in the Ising model and the ANNNI model in transverse field using the interface approach. The exact result of the Ising chain in a transverse field is reproduced. We find that apart from the interfacial energy, there are two other response functions which show simple scaling behaviour. For the ANNNI model in a transverse field, the phase diagram can be fully studied in the region where a ferromagnetic to paramagnetic phase transition occurs. The other region can be studied partially; the boundary where the antiphase vanishes can be estimated.Comment: 11 pages, Revtex, 9 figures To be published in Physical Reveiw B , May 199

Realistic searches on stretched exponential networks

We consider navigation or search schemes on networks which have a degree distribution of the form $P(k) \propto \exp(-k^\gamma)$. In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter $\lambda$. The searches are realistic in the sense that not all search chains can be completed. An estimate of $\mu=\rho/s_d$, where $\rho$ is the success rate and $s_d$ the dynamic path length, shows that for a network of $N$ nodes, $\mu \propto N^{-\delta}$ in general. Dynamic small world effect, i.e., $\delta \simeq 0$ is shown to exist in a restricted region of the $\lambda-\gamma$ plane.Comment: Based on talk given in Statphys Guwahati, 200

Non-local conservation in the coupling field: effect on critical dynamics

We consider the critical dynamics of a system with an $n$-component non-conserved order parameter coupled to a conserved field with long range diffusion. An exponent $\sigma$ characterizes the long range transport, $\sigma=2$ being the known locally conserved case. With renormalisation group calculations done upto one loop order, several regions are found with different values of the dynamic exponent $z$ in the $\sigma -n$ plane. For $n<4$, there are three regimes, I: nonuniversal, $\sigma$ dependent $z$, II: universal with $z$ depending on $n$ and III': conservation law irrelevant, $z$ being equal to that in the nonconserved case. The known locally conserved case belongs to regions I and II.Comment: 4 pages, revtex, 1 eps figure included, to appear in Journal of Physics

Continuous utility factor in segregation models

We consider the constrained Schelling model of social segregation in which the utility factor of agents strictly increases and non-local jumps of the agents are allowed. In the present study, the utility factor u is defined in a way such that it can take continuous values and depends on the tolerance threshold as well as the fraction of unlike neighbours. Two models are proposed: in model A the jump probability is determined by the sign of u only which makes it equivalent to the discrete model. In model B the actual values of u are considered. Model A and model B are shown to differ drastically as far as segregation behaviour and phase transitions are concerned. In model A, although segregation can be achieved, the cluster sizes are rather small. Also, a frozen state is obtained in which steady states comprise of many unsatisfied agents. In model B, segregated states with much larger cluster sizes are obtained. The correlation function is calculated to show quantitatively that larger clusters occur in model B. Moreover for model B, no frozen states exist even for very low dilution and small tolerance parameter. This is in contrast to the unconstrained discrete model considered earlier where agents can move even when utility remains same. In addition, we also consider a few other dynamical aspects which have not been studied in segregation models earlier.Comment: 9 pages, 17 figure

Zero temperature coarsening in Ising model with asymmetric second neighbour interaction in two dimensions

We consider the zero temperature coarsening in the Ising model in two dimensions where the spins interact within the Moore neighbourhood. The Hamiltonian is given by $H = - \sum_{}{S_iS_j} - \kappa \sum_{}{S_iS_{j'}}$ where the two terms are for the first neighbours and second neighbours respectively and $\kappa \geq 0$. The freezing phenomena, already noted in two dimensions for $\kappa=0$, is seen to be present for any $\kappa$. However, the frozen states show more complicated structure as $\kappa$ is increased; e.g. local anti-ferromagnetic motifs can exist for $\kappa>2$. Finite sized systems also show the existence of an iso-energetic active phase for $\kappa > 2$, which vanishes in the thermodynamic limit. The persistence probability shows universal behaviour for $\kappa>0$, however it is clearly different from the $\kappa=0$ results when non-homogeneous initial condition is considered. Exit probability shows universal behaviour for all $\kappa \geq 0$. The results are compared with other models in two dimensions having interactions beyond the first neighbour.Comment: 8 pages, 12 figure

Persistence of unvisited sites in presence of a quantum random walker

A study of persistence dynamics is made for the first time in a quantum system by considering the dynamics of a quantum random walk. For a discrete walk on a line starting at $x=0$ at time $t=0$, the persistence probability $P(x,t)$ that a site at $x$ has not been visited till time $t$ has been calculated. $P(x,t)$ behaves as $(t/|x|-1)^{-\alpha}$ with $\alpha \sim 0.3$ while the global fraction ${\cal{P}}(t) = \sum_xP(x,t)/2t$ of sites remaining unvisited at time $t$ attains a constant value. $F(x,t)$, the probability that the site at $x$ is visited for the first time at $t$ behaves as $(t/|x|-1)^{-\beta}/|x|$ where $\beta = 1+ \alpha$ for $t/|x|>> 1$,and ${\cal{F}}(t) =\sum_xF(x,t)/2t \sim 1/t$. A few other properties related to the persistence and first passage times are studied and some fundamental differences between the classical and the quantum cases are observed.Comment: 5 pages, 6 figures, revtex