First passage time in a two-layer system

As a first step in the first passage problem for passive tracer in stratified porous media, we consider the case of a two-dimensional system consisting of two layers with different convection velocities. Using a lattice generating function formalism and a variety of analytic and numerical techniques, we calculate the asymptotic behavior of the first passage time probability distribution. We show analytically that the asymptotic distribution is a simple exponential in time for any choice of the velocities. The decay constant is given in terms of the largest eigenvalue of an operator related to a half-space Green's function. For the anti-symmetric case of opposite velocities in the layers, we show that the decay constant for system lengthL crosses over fromL ^–2 behavior in the diffusive limit toL ^–1 behavior in the convective regime, where the crossover lengthL ^* is given in terms of the velocities. We also have formulated a general self-consistency relation, from which we have developed a recursive approach which is useful for studying the short-time behavior.     

Comment on a paper by G. H. Weiss,S. Havlin,and A. Bunde

A simple proof is pointed out for the asymptotic exponential decay of then-step survival probability of a random walk on a finite lattice with traps in the limit asn . Some bounds are mentioned, which are valid for finiten and for symmetric random walks.     

Random walks in a random field of decaying traps

We study random walks on ^d (d 1) containing traps subject to decay. The initial trap distribution is random. In the course of time, traps decay independently according to a given lifetime distribution. We derive a necessary and sufficient condition under which the walk eventually gets trapped with probability 1. We prove bounds and asymptotic estimates for the survival probability as a function of time and for the average trapping time. These are compared with some well-known results for nondecaying traps.     

The Constants in the Central Limit Theorem for the One-Dimensional Edwards Model

The Edwards model in one dimension is a transformed path measure for one-dimensional Brownian motion discouraging self-intersections. We study the constants appearing in the central limit theorem (CLT) for the endpoint of the path (which represent the mean and the variance) and the exponential rate of the normalizing constant. The same constants appear in the weak-interaction limit of the one-dimensional Domb–Joyce model. The Domb–Joyce model is the discrete analogue of the Edwards model based on simple random walk, where each self-intersection of the random walk path recieves a penalty e ^–2. We prove that the variance is strictly smaller than 1, which shows that the weak interaction limits of the variances in both CLTs are singular. The proofs are based on bounds for the eigenvalues of a certain one-parameter family of Sturm–Liouville differential operators, obtained by using monotonicity of the zeros of the eigen-functions in combination with computer plots.     

Random walks on lattices with randomly distributed traps I. The average number of steps until trapping

For a random walk on a lattice with a random distribution of traps we derive an asymptotic expansion valid for smallq for the average number of steps until trapping, whereq is the probability that a lattice point is a trap. We study the case of perfect traps (where the walk comes to an end) and the extension obtained by letting the traps be imperfect (i.e., by giving the walker a finite probability to remain free when stepping on a trap). Several classes of random walks of varying dimensionality are considered and special care is taken to show that the expansion derived is exact up to and including the last term calculated. The numerical accuracy of the expansion is discussed.     

Long- and Short-Range Correlations in Genome Organization

We study the size distribution of coding and non-coding regions in DNA sequences. For most organisms we observe that the size distribution P _c(S) of the coding regions of size S shows short range distribution, whereas the size distribution of the non-coding regions follows a power-law decay P _nc(S) S ^{–1 –} , with power exponents indicating clear long-range behavior. We argue, using the Generalized Central Limit Theorem, that the long-range distributions observed in the non-coding are related to the lower level clustering of purines and pyrimidines (1d islands) which follow similar long-range laws. We also address the question of clustering of coding segments in the two complementary strands of DNA. We observe a short-range clustering of coding regions in both strands, expressed by an exponential decay in the clustering size distribution. The decay exponent expresses the degree of short-range correlations and the deviation from random clustering.     

Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty

We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[–|t–s| ^–p ] wheret ands are the times at which the common site is visited andp is a parameter. We prove that ifp<1 and is sufficiently large, then the walk behaves ballistically, i.e., the distance to the end of the walk grows linearly with the number of steps in the walk. We also give a heuristic argument that ifp>3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation.     

Survival of Classical and Quantum Particles in the Presence of Traps

We present a detailed comparison of the motion of a classical and of a quantum particle in the presence of trapping sites, within the framework of continuous-time classical and quantum random walk. The main emphasis is on the qualitative differences in the temporal behavior of the survival probabilities of both kinds of particles. As a general rule, static traps are far less efficient to absorb quantum particles than classical ones. Several lattice geometries are successively considered: an infinite chain with a single trap, a finite ring with a single trap, a finite ring with several traps, and an infinite chain and a higher-dimensional lattice with a random distribution of traps with a given density. For the latter disordered systems, the classical and the quantum survival probabilities obey a stretched exponential asymptotic decay, albeit with different exponents. These results confirm earlier predictions, and the corresponding amplitudes are evaluated. In the one-dimensional geometry of the infinite chain, we obtain a full analytical prediction for the amplitude of the quantum problem, including its dependence on the trap density and strength.     

Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate γ, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P(c,t)∼exp(−Ct^1/2). The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.     

Self-organization of aging in a population approaching the steady state

The nonequilibrium asymptotic dynamics of a model for aging in a population of individuals initially having a random distribution of survival rates is studied. The model drives itself toward a steady state, and the average age tends toward a well-defined value. An analytic derivation shows that the average age of the members of the population decays in a power law fashion with the leading term of ordert ^–1. Monte Carlo simulations agree with the analytic work, and show that thet ^–1 decay is universally observed even when somatic mutations are introduced into the population.     

On the Anisotropic Walk on the Supercritical Percolation Cluster

We investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on ^d, d2. In particular we show that for small anisotropy the walk behaves in a ballistic fashion, whereas for strong anisotropy the walk is sub-diffusive. For arbitrary anisotropy, we also prove the directional transience of the walk and construct a renewal structure.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

One-dimensional non-nearest-neighbor random walks in the presence of traps

A one-dimensional lattice random walk in the presence ofm equally spaced traps is considered. The step length distribution is a symmetric exponential. An explicit analytic expression is obtained for the probability that the random walk will be trapped at thejth trapping site.     

Force Relaxation in the q-Model for Granular Media

We study the relaxation of force distributions in the q-model, assuming a uniform q-distribution. We show that diffusion of correlations makes this relaxation very slow. On a d-dimensional lattice, the asymptotic state is approached as l ^(1–d)/2, where l is the number of layers from the top. Furthermore, we derive asymptotic modes of decay, along which an arbitrary short-range correlated initial distribution will decay towards the stationary state.     

Freezing of nonequilibrium domain structures in a kinetic Ising model

We study the freezing of a disordered spin structure upon continuous cooling to absolute zero for a kinetic Ising spin chain with alternating weak and strong bonds. The kinetic equation for the spin pair correlation function is solved analytically in a continuum approximation. The exponent for the asymptotic dependence of the frozen kink density on a characteristic cooling time is found to bez ^–1, wherez is the equilibrium dynamic critical exponent, for a universality class including power-law and exponential cooling, and 1/2 for a logarithmic cooling program which exhibits threshold behavior.     

An exponential upper bound for the survival probability in a dynamic random trap model

We consider a symmetric translation-invariant random walk on thed-dimensional lattice ? ^d . The walker moves in an environment of moving traps. When the walker hits a trap, he is killed. The configuration of traps in the course of time is a reversible Markov process satisfying a level-2 large-deviation principle. Under some restrictions on the entropy function, we prove an exponential upper bound for the survival probability, i.e., $$\mathop {lim sup}\limits_{t \to \infty } \frac{1}{t}\log \mathbb{P}(T \geqslant t)< 0$$ whereT is the survival time of the walker. As an example, our results apply to a random walk in an environment of traps that perform a simple symmetric exclusion process.     

The broken supersymmetry phase of a self-avoiding random walk

We consider a weakly self-avoiding random walk on a hierarchical lattice ind=4 dimensions. We show that for choices of the killing ratea less than the critical valuea _cthe dominant walks fill space, which corresponds to a spontaneously broken supersymmetry phase. We identify the asymptotic density to which walks fill space, (a), to be a supersymmetric order parameter for this transition. We prove that (a)(a _c–a) (–log(a _c–a))^1/2 asaa _c, which is mean-field behavior with logarithmic corrections, as expected for a system in its upper critical dimension.Research partially supported by NSF Grants DMS 91-2096 and DMS 91-96161.     

Energy transfer as a random walk with long-range steps

We consider the incoherent energy transport in molecular crystals, where the transfer rates stem from Coulombic and exchange interactions. For substitutionally disordered lattices we present in a first passage model the excitation decay due to trapping by randomly distributed traps; the decay is related to the distribution of the number of distinct sites visited during the timet and is expressible through the cumulants of this distribution. The validity domains of approximate decay laws based on the first few cumulants are also discussed. We exemplify the findings for dipolar transfer rates between randomly distributed molecules on a square lattice, by comparing the random walk on the random system to its CTRW (continuous time random walk) counterpart.     

Algebraic spinors and directed random walks in the McKane-Parisi-Sourlas theorem

We present the Dirac propagator as a random walk on anS ^D–1 sphere for Majorana spinors, even spinor space, Dirac spinors, and Chevalley-Crumeyrolle spinors built from Minkowski space. We propose the Dirac propagator constructed from Chevalley-Crumeyrolle spinors as the generators of a Markov process such that McKane-Parisi-Sourlas theorem can be applied to calculate the expectation values for functions of local times.     

Lifshitz Tails in Constant Magnetic Fields

We consider the 2D Landau Hamiltonian H perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of H. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, exponential sub-Gaussian, and super-Gaussian decay of the one-site potential. If the edge is away from the Landau levels, we impose a rational-flux assumption on the magnetic field, consider compactly supported one-site potentials, and formulate a theorem which is analogous to a result obtained by the first author and T. Wolff in [25] for the case of a vanishing magnetic field.