Harmonically Confined Particles with Long-Range Repulsive Interactions

We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that behaves as a power law ∝∑[under i≠j][over N]|x_{i}-x_{j}|^{-k} (with k>-2) of their mutual distance. This is a generalization of the well-known cases of the one-component plasma (k=-1), Dyson's log gas (k→0^{+}), and the Calogero-Moser model (k=2). Because of the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all k>-2. We compute exactly the average density profile for large N for all k>-2 and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on k with distinct behavior for -2<k<1, k>1 and k=1.

Instanton approach to large N Harish-Chandra-Itzykson-Zuber integrals

We reconsider the large N asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson's Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin's equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.

Persistence in a stationary time series

We study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time. We show that under certain conditions, the persistence of such a continuous process reduces to the persistence of a corresponding discrete sequence obtained from the measurement of the process only at integer times. We then construct a specific sequence for which the persistence can be computed even though the sequence is non-Markovian. We show that this may be considered as a limiting case of persistence in the diffusion process on a hierarchical lattice.

Fluctuation-dominated phase ordering driven by stochastically evolving surfaces: depth models and sliding particles

We study an unconventional phase ordering phenomenon in coarse-grained depth models of the hill-valley profile of fluctuating surfaces with zero overall tilt, and for hard-core particles sliding on such surfaces under gravity. We find that several such systems approach an ordered state with large scale fluctuations which make them qualitatively different from conventional phase ordered states. We consider surfaces in the Edwards-Wilkinson (EW), Kardar-Parisi-Zhang (KPZ) and noisy surface-diffusion (NSD) universality classes. For EW and KPZ surfaces, coarse-grained depth models of the surface profile exhibit coarsening to an ordered steady state in which the order parameter has a broad distribution even in the thermodynamic limit, the distribution of particle cluster sizes decays as a power-law (with an exponent straight theta), and the scaled two-point spatial correlation function has a cusp (with an exponent alpha=1/2) at small values of the argument. The latter feature indicates a deviation from the Porod law which holds customarily, in coarsening with scalar order parameters. We present several numerical and exact analytical results for the coarsening process and the steady state. For linear surface models with a dynamical exponent z, we show that alpha=(z-1)/2 for z<3 and alpha=1 for z>3, and there are logarithmic corrections for z=3, implying alpha=1/2 for the EW surface and 1 for the NSD surface. Within the independent interval approximation we show that alpha+straight theta=2. We also study the dynamics of hard-core particles sliding locally downward on these fluctuating one-dimensional surfaces, and find that the surface fluctuations lead to large-scale clustering of the particles. We find a surface-fluctuation driven coarsening of initially randomly arranged particles; the coarsening length scale grows as approximately t(1/z). The scaled density-density correlation function of the sliding particles shows a cusp with exponents alpha approximately 0.5 and 0.25 for the EW and KPZ surfaces. The particles on the NSD surface show conventional coarsening (Porod) behavior with alpha approximately 1.

Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence

We study analytically the distribution of the minimum of a set of hierarchically correlated random variables E1, E2,ellipsis, E(N) where E(i) represents the energy of the ith path of a directed polymer on a Cayley tree. If the variables were uncorrelated, the minimum energy would have an asymptotic Gumbel distribution. We show that due to the hierarchical correlations, the forward tail of the distribution of the minimum energy becomes highly nonuniversal, depends explicitly on the distribution of the bond energies epsilon, and is generically different from the superexponential forward tail of the Gumbel distribution. The consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistence is also shown to be generically anomalous.

Extremal properties of random trees

We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights approaches a traveling wave form. The wave front in the minimal case is governed by the small-extremal-height tail of the distribution, and conversely, the front in the maximal case is governed by the large-extremal-height tail of the distribution. We determine several statistical characteristics of the extremal height distribution analytically. In particular, the expected minimal and maximal heights grow logarithmically with the tree size, N, h(min) approximately v(min) ln N, and h(max) approximately v(max) ln N, with v(min)=0.373365ellipsis and v(max)=4.31107ellipsis, respectively. Corrections to this asymptotic behavior are of order O(ln ln N).

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 autofluctuation of a tracer particle grows subdiffusively sigma(2)0(t) approximately t(1/2) for large time t in the absence of external bias but grows diffusively sigma(2)0(t) approximately 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(2)(r)(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 on the time dependence of sigma(2)(r)(t). For fixed r,sigma(2)(r)(t) increases monotonically with t in the absence of bias, but has a nonmonotonic dependence on t in the presence of bias. Similarities and differences with the simple exclusion process are also discussed.

Persistence of a continuous stochastic process with discrete-time sampling

We introduce the concept of "discrete-time persistence," which deals with zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T=n Delta T. For a Gaussian Markov process with relaxation rate mu, we show that the persistence (no crossing) probability decays as [rho(a)](n) for large n, where a = exp(-mu Delta T), and we compute rho(a) to high precision. We also define the concept of "alternating persistence," which corresponds to a<0. For a>1, corresponding to motion in an unstable potential (mu<0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.

Spatial persistence of fluctuating interfaces

We show that the probability, P0(l), that the height of a fluctuating (d+1)-dimensional interface in its steady state stays above its initial value up to a distance l, along any linear cut in the d-dimensional space, decays as P0(l) approximately l(theta). Here straight theta is a "spatial" persistence exponent, and takes different values, straight theta(s) or straight theta(0), depending on how the point from which l is measured is specified. These exponents are shown to map onto corresponding temporal persistence exponents for a generalized d = 1 random-walk equation. The exponent straight theta(0) is nontrivial even for Gaussian interfaces.

Dynamics of efficiency: a simple model

We propose a simple model that describes the dynamics of efficiencies of competing agents. Agents communicate leading to increase of efficiencies of underachievers, and an efficiency of each agent can increase or decrease irrespectively of other agents. When the rate of deleterious changes exceeds a certain threshold, the system falls into a stagnant phase. In the opposite situation, the average efficiency improves with asymptotically constant rate and the efficiency distribution has a finite width. The leading algebraic corrections to the asymptotic growth rate are also computed.

Coarsening in the presence of kinetic disorders: analogy to granular compaction

We study the zero temperature dynamics in an Ising chain in the presence of a dynamically induced field that favors locally the " -" phase compared to the " +" phase. At late times, while the " +" domains coarsen as t(1/2), the " -" domains coarsen as t(1/2)log(t). Hence, at late times, the magnetization decays slowly as m(t) = -1+const/log(t). We establish this behavior both analytically within an independent interval approximation and numerically. Our model can be viewed as a simple model for granular compaction, where the system decays into a fully compact state (with all spins " -") in a slow logarithmic manner as seen in recent experiments on granular systems.

Exact phase diagram of a model with aggregation and chipping

We reexamine a simple lattice model of aggregation in which masses diffuse and coalesce upon contact with rate 1 and every nonzero mass chips off a single unit of mass and adds it to a randomly chosen neighbor with rate w. The dynamics conserves the average mass density rho and in the stationary state the system undergoes a nonequilibrium phase transition in the (rho-w) plane across a critical line rho(c)(w). In this paper, we show analytically that in arbitrary spatial dimensions rho(c)(w)=sqrt[w+1]-1 exactly and hence, remarkably, is independent of dimension. We also provide direct and indirect numerical evidence that strongly suggests that the mean field asymptotic results for the single site mass distribution function and the associated critical exponents are superuniversal, i.e., independent of dimension.

Traveling waves, front selection, and exact nontrivial exponents in a random fragmentation problem

We study a random bisection problem where an interval of length x is cut into two random fragments at the first stage, then each of these two fragments is cut further, etc. We compute the probability P(n)(x) that at the nth stage, each of 2(n) fragments is shorter than 1. We show that P(n)(x) approaches a traveling wave form, and the front position x(n) increases as x(n) approximately n(beta)rho(n) for large n with rho = 1.261 076ellipsis and beta = 0.453 025ellipsis. We also solve the m-section problem where each interval is broken into m fragments and show that rho(m) approximately m/(lnm) and beta(m) approximately 3/(2lnm) for large m. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.

Extremal paths on a random cayley tree

We investigate the statistics of extremal path(s) (both the shortest and the longest) from the root to the bottom of a Cayley tree. The lengths of the edges are assumed to be independent identically distributed random variables drawn from a distribution rho(l). Besides, the number of branches from any node is also random. Exact results are derived for arbitrary distribution rho(l). In particular, for the binary 0,1 distribution rho(l)=pdelta(l,1)+(1-p)delta(l, 0), we show that as p increases, the minimal length undergoes an unbinding transition from a "localized" phase to a "moving" phase at the critical value, p=p(c)=1-b(-1), where b is the average branch number of the tree. As the height n of the tree increases, the minimal length saturates to a finite constant in the localized phase (p<p(c)), but increases linearly as v(min)(p)n in the moving phase (p>p(c)) where the velocity v(min)(p) is determined via a front selection mechanism. At p=p(c), the minimal length grows with n in an extremely slow double-logarithmic fashion. The length of the maximal path, on the other hand, increases linearly as v(max)(p)n for all p. The maximal and minimal velocities satisfy a general duality relation, v(min)(p)+v(max)(1-p)=1, which is also valid for directed paths on finite-dimensional lattices.

Exact calculation of the spatiotemporal 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 spatiotemporal 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 spatiotemporal force-force correlation function in the q model. These can be used to test the applicability of the q model in experiments.

Phase transition in the takayasu model with desorption

We study a lattice model where particles carrying different masses diffuse and coalesce upon contact, and also unit masses adsorb to a site with rate q or desorb from a site with nonzero mass with rate p. In the limit p=0 (without desorption), our model reduces to the well studied Takayasu model where the steady-state single site mass distribution has a power-law tail P(m) approximately m(-tau) for large mass. We show that varying the desorption rate p induces a nonequilibrium phase transition in all dimensions. For fixed q, there is a critical p(c)(q) such that if p<p(c)(q), the steady-state mass distribution, P(m) approximately m(-tau) for large m as in the Takayasu case. For p=p(c)(q), we find P(m) approximately m(-tau(c)) where tau(c) is a new exponent, while for p>p(c)(q), P(m) approximately exp(-m/m(*)) for large m. The model is studied analytically within a mean-field theory and numerically in one dimension.

Analytical results for random walk persistence

In this paper, we present a detailed calculation of the persistence exponent straight theta for a nearly Markovian Gaussian process X(t), a problem initially introduced elsewhere in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. Resummed perturbative and nonperturbative expressions for straight theta are derived, which suggest a connection with the result of the alternative independent interval approximation. The perturbation theory is extended to the calculation of straight theta for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and nonperturbative expressions for the persistence exponent straight theta(X0), describing the probability that the process remains larger than X(0)sqrt[<X2(t)>].

Residence time distribution for a class of Gaussian Markov processes

We study the distribution of residence time or equivalently that of "mean magnetization" for a family of Gaussian Markov processes indexed by a positive parameter alpha. The persistence exponent for these processes is simply given by theta=alpha but the residence time distribution is nontrivial. The shape of this distribution undergoes a qualitative change as theta increases, indicating a sharp change in the ergodic properties of the process. We develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha. For some special values of alpha, we obtain closed form expressions of the distribution function.