Persistence of Randomly Coupled Fluctuating Interfaces

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h_1 and h_2 respectively, each growing over a d-dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h_2, however, is coupled to h_1 via a quenched random velocity field. In the limit d\to 0, our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t_0\to \infty, the stochastic process h_2, at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H_2=1-\beta_1/2, where \beta_1 is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be \theta_s^2=1-H_2=\beta_1/2. These analytical results are verified by numerical simulations.Comment: 15 pages, 3 .eps figures include

Persistence and the Random Bond Ising Model in Two Dimensions

We study the zero-temperature persistence phenomenon in the random bond $\pm J$ Ising model on a square lattice via extensive numerical simulations. We find strong evidence for ` blocking\rq regardless of the amount disorder present in the system. The fraction of spins which {\it never} flips displays interesting non-monotonic, double-humped behaviour as the concentration of ferromagnetic bonds $p$ is varied from zero to one. The peak is identified with the onset of the zero-temperature spin glass transition in the model. The residual persistence is found to decay algebraically and the persistence exponent $\theta (p)\approx 0.9$ over the range $0.1\le p\le 0.9$. Our results are completely consistent with the result of Gandolfi, Newman and Stein for infinite systems that this model has ` mixed\rq behaviour, namely positive fractions of spins that flip finitely and infinitely often, respectively. [Gandolfi, Newman and Stein, Commun. Math. Phys. {\bf 214} 373, (2000).]Comment: 9 pages, 5 figure

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the `sampling interval' used in the measurement for both `steady-state' and `finite' initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A `deterministic approximation' is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.Comment: RevTeX4, 21 pages, 8 .eps figures, changes in sections IIIB and IIIC and in Figs 7 and 8, version to be published in Physical Review

Understanding Search Trees via Statistical Physics

We study the random m-ary search tree model (where m stands for the number of branches of a search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a traveling front structure. In addition, the variance of the number of nodes needed to store a data string of a given size N is shown to undergo a striking phase transition at a critical value of the branching ratio m_c=26. We identify the mechanism of this phase transition, show that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the proceedings of STATPHYS-2

Exact Calculation of the Spatio-temporal Correlations in the Takayasu model and in the q-model of Force Fluctuations in Bead Packs

We calculate exactly the two point mass-mass correlations in arbitrary spatial dimensions in the aggregation model of Takayasu. In this model, masses diffuse on a lattice, coalesce upon contact and adsorb unit mass from outside at a constant rate. Our exact calculation of the variance of mass at a given site proves explicitly, without making any assumption of scaling, that the upper critical dimension of the model is 2. We also extend our method to calculate the spatio-temporal correlations in a generalized class of models with aggregation, fragmentation and injection which include, in particular, the $q$-model of force fluctuations in bead packs. We present explicit expressions for the spatio-temporal force-force correlation function in the $q$-model. These can be used to test the applicability of the $q$-model in experiments.Comment: 15 pages, RevTex, 2 figure

Exact Tagged Particle Correlations in the Random Average Process

We study analytically the correlations between the positions of tagged particles in the random average process, an interacting particle system in one dimension. We show that in the steady state the mean squared auto-fluctuation of a tracer particle grows subdiffusively as $sigma^2(t) ~ t^{1/2}$ for large time t in the absence of external bias, but grows diffusively $sigma^2(t) ~ t$ in the presence of a nonzero bias. The prefactors of the subdiffusive and diffusive growths as well as the universal scaling function describing the crossover between them are computed exactly. We also compute $sigma_r^2(t)$, the mean squared fluctuation in the position difference of two tagged particles separated by a fixed tag shift r in the steady state and show that the external bias has a dramatic effect in the time dependence of $sigma_r^2(t)$. For fixed r, $sigma_r^2(t)$ increases monotonically with t in absence of bias but has a non-monotonic dependence on t in presence of bias. Similarities and differences with the simple exclusion process are also discussed.Comment: 10 pages, 2 figures, revte

Self-organisation to criticality in a system without conservation law

We numerically investigate the approach to the stationary state in the nonconservative Olami-Feder-Christensen (OFC) model for earthquakes. Starting from initially random configurations, we monitor the average earthquake size in different portions of the system as a function of time (the time is defined as the input energy per site in the system). We find that the process of self-organisation develops from the boundaries of the system and it is controlled by a dynamical critical exponent z~1.3 that appears to be universal over a range of dissipation levels of the local dynamics. We show moreover that the transient time of the system $t_{tr}$ scales with system size L as $t_{tr} \sim L^z$. We argue that the (non-trivial) scaling of the transient time in the OFC model is associated to the establishment of long-range spatial correlations in the steady state.Comment: 10 pages, 6 figures; accepted for publication in Journal of Physics