Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models

We compute exactly the distribution of the occupation time in a discrete {\em non-Markovian} toy sequence which appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function which is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting new question for a generic finite sized spin glass model: at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.Comment: 10 pages Revtex (two-column), 1 eps figure (included

Survival in equilibrium step fluctuations

We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical STM data on step fluctuations on Al/Si(111) and Ag(111) surfaces.Comment: RevTeX, 4 pages, 3 figure

Persistence of a Continuous Stochastic Process with Discrete-Time Sampling: Non-Markov Processes

We consider the problem of `discrete-time persistence', which deals with the zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T = n(\Delta T). For a Gaussian Stationary Process the persistence (no crossing) probability decays as exp(-\theta_D T) = [\rho(a)]^n for large n, where a = \exp[-(\Delta T)/2], and the discrete persistence exponent, \theta_D, is given by \theta_D = \ln(\rho)/2\ln(a). Using the `Independent Interval Approximation', we show how \theta_D varies with (\Delta T) for small (\Delta T) and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker. We extend the matrix method developed by us previously [Phys. Rev. E 64, 015151(R) (2001)] to determine \rho(a) for a two-dimensional random walk and the one-dimensional random acceleration problem. We also consider `alternating persistence', which corresponds to a < 0, and calculate \rho(a) for this case.Comment: 14 pages plus 8 figure

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d) > 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n \gg 1 even, the probability that they have no real root on the full real axis decays like n^{-2(\theta(2)+\theta(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like \exp{(-2\theta_{\infty}} \sqrt{n}) and \exp{(-\pi \theta_{\infty} \sqrt{n})} where \theta_{\infty} is such that \theta(d) = 2^{-3/2} \theta_{\infty} \sqrt{d} in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde \phi(k/N_{ab}))} where N_{ab} is the mean number of real roots in [a,b] and \tilde \phi(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent {-2}. These analytical results are confirmed by detailed numerical computations.Comment: 32 pages, 16 figure

DISTRIBUTION AND SYNAPTIC ORGANIZATION OF SEROTONINERGIC AND NORADRENERGIC AXONS IN THE LATERAL GENICULATE-NUCLEUS OF THE RAT

Contact processes describe the transmission of distinct properties of nodes via the links of a network. They provide a simple framework for many phenomena, such as epidemic spreading and opinion formation. Combining contact processes with rules for topological evolution yields an adaptive network in which the states of the nodes can interact dynamically with the topological degrees of freedom. By moment-closure approximation it is possible to derive low-dimensional systems of ordinary differential equations that describe the dynamics of the adaptive network on a coarse-grained level. In this chapter we discuss the approximation technique itself as well as its applications to adaptive networks. Thus, it can serve both as a tutorial as well as a review of recent results.Comment: 18 pages, 5 figure

Network evolution based on minority game with herding behavior

The minority game (MG) is used as a source of information to design complex networks where the nodes represent the playing agents. Differently from classical MG consisting of independent agents, the current model rules that connections between nodes are dynamically inserted or removed from the network according to the most recent game outputs. This way, preferential attachment based on the concept of social distance is controlled by the agents wealth. The time evolution of the network topology, quantitatively measured by usual parameters, is characterized by a transient phase followed by a steady state, where the network properties remain constant. Changes in the local landscapes around individual nodes depend on the parameters used to control network links. If agents are allowed to access the strategies of their network neighbors, a feedback effect on the network structure and game outputs is observed. Such effect, known as herding behavior, considerably changes the dependence of volatility σ on memory size: it is shown that the absolute value of σ as well as the corresponding value of memory size depend both on the network topology and on the way along which the agents make their playing decisions in each game round