Anomalous fluctuations in random walk dynamics

The anomalous dispersion of noninteracting particles randomly walking in a network is considered. It is shown that the existence of large dangling branches attached to a backbone induces a l/f-like behavior in the current autocorrelation function at low frequencies. The waiting times associated with dangling loops scale liket ^–3/2. The size of the dangling branches provides a lower cutoff to the power law behavior. When the side branches are infinite, self-similar structures, the power law behavior persists up to a zero frequency. The currents we consider are created either by a bias on the random walk or by a current source. We consider both the total current, which is often referred to in the literature, and the current measured at endpoints of a specimen attached to a (model) battery. The differences and similarities between the two corresponding correlations are analyzed. In particular, we find that in the second case l/f noise exists only for large bias. When a statistical distribution of dangling branches is considered, we find that the largest power of frequency in the spectrum is 1.13. Much of our results are true when the dangling branches are replaced by traps having waiting time distributions that equal those of the branches. The waiting time associated with a power law distribution of dangling loops (m ^–x:m is the length of the loop) scales liket ^–1 –(x/2). However, it is shown that geometry alone can be responsible for the appearance of power laws in the spectra. Random geometry can be regarded as a model (or source) of random hopping times.     

DNA unzipping and the unbinding of directed polymers in a random media

We consider the unbinding of a directed polymer in a random media from a wall in d=1+1 dimensions and a simple one-dimensional model for DNA unzipping. Using the replica trick we show that the restricted partition functions of these problems are identical up to an overall normalization factor. Our finding gives an example of a generalization of the stochastic matrix form decomposition to disordered systems, a method which allows us to reduce the dimensionality of the problem. The equivalence between the two problems, for example, allows us to derive the probability distribution for finding the directed polymer a distance z from the wall. We discuss implications of these results for the related Kardar-Parisi-Zhang equation and the asymmetric exclusion process.     

Diffusion of directed polymers in a random environment

We consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time. In spatial dimensionsd>2, we establish that the behavior is diffusive with probability one. The diffusion constant is not renormalized by the interaction.     

Anomalous transport fluctuations in a model of irregular media

We consider a diffusion model with stochastic porosity for which the average solution exhibits an abnormal transport. In this paper we investigate the relation of such an anomalous diffusive property of the mean value with the behavior of the solution corresponding to each realization of the stochastic porosity. Such a solution will correspond to the actual measurements in an experiment made on a particular tube. The most relevant result of our work is that, although the concentration corresponding to each realization diffuses normally for large times, it experiments on large deviations from the mean value during intermediate times.     

Diffusion of directed polymers in a strong random environment

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. It was shown by Imbrie and Spencer that in spatial dimensions three or above the behavior is diffusive if the directed polymer interacts weakly with the environment and if the random environment follows the Bernoulli distribution. Under the same assumption on the random environment as that of Imbrie and Spencer, we establish that in spatial dimensions four or above the behavior is still diffusive even when the directed polymer interacts strongly with the environment. More generally, we can prove that, if the random environment is bounded and if the supremum of the support of the distribution has a positive mass, then there is an integerd ₀ such that in dimensions higher thand ₀ the behavior of the random polymer is always diffusive.     

On the Bethe ansatz for random directed polymers

We show that the inclusion of the (gapless) center-of-mass motion together with a functional integral representation of the Bethe wave function allows one to predict exactly the critical exponents for random directed polymers in (1+1) dimensions. The corresponding amplitudes are computed; they compare satisfactorily with existing numerical data. Within a replica-symmetric theory, we find that the Green function of the polymer has the form recently proposed by Parisi.     

A remark on diffusion of directed polymers in random environments

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one ifd>2 and <₀, where ₀ is defined in terms of the probability that the symmetric nearest neighbor random walk on thed-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.     

Anomalous polarization effects during light scattering in random media

The dependence of the intensity of light backscattered from a layer of a randomly inhomogeneous medium on the polarization of incident light and the size of scatterers has been investigated. The results of numerical simulation have demonstrated that the direction of rotation of the plane of polarization is different in systems with small- and large-scale inhomogeneities. It is shown for the first time that the dependence of the sign of the residual circular polarization on the size of scatterers can be observed in systems described by the Henyey-Greenstein phase function used in simulating biological tissues. A similar anomalous polarization effect, which consists in changing the direction of rotation of the plane of polarization of backscattered light with an increase in the scattering angle, is revealed in studying the coherent backscattering component. These polarization effects are observed in light backscattering from optically active media.     

Sign phase transition and directed paths in random media

We consider a lattice model which corresponds to the high temperature expansions of disordered Ising and Heisenberg models and to the deeply localized regime of the disordered Anderson model. The spin correlation functions for the Ising and Heisenberg model and the amplitude of electron tunneling for the Anderson model exhibit a “sign phase transition.” At small concentration x of scatterers with a negative scattering amplitude these quantities have predictable signs while at large x their signs are unpredictable. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 4, 283–288 (25 August 1996) Published in English in the original Russian Journal. Edited by J. R. Anderson.     

Path integrals for wave intensity fluctuations in random media

Approximate expressions for the fourth order moment of a wave propagating in a random medium are derived by using the path integral formulation. These solutions allow the spectrum of intensity fluctuations of a multiply scattered wave to be found, and they are valid at all distances in the medium. The results obtained by path integral methods turn out to be the same as those obtained previously by solving the parabolic partial differential equation for the fourth moment. The spatial frequency spectra of intensity fluctuations are evaluated for a medium in which the irregularities have a single scale and also for one in which there is a range of scale sizes.     

The largest Liapunov exponent for random matrices and directed polymers in a random environment

We study the largest Liapunov exponent for products of random matrices. The two classes of matrices considered are discrete,d-dimensional Laplacians, with random entries, and symplectic matrices that arise in the study ofd-dimensional lattices of coupled, nonlinear oscillators. We derive bounds on this exponent for all dimensions,d, and we show that ifd3, and the randomness is not too strong, one can obtain an explicit formula for the largest exponent in the thermodynamic limit. Our method is based on an equivalence between this problem and the problem of directed polymers in a random environment.     

Mean field theory of directed polymers with random complex weights

We show that for the problem of directed polymers on a tree with i.i.d. random complex weights on each bond, three possible phases can exist; the phase of a particular system is determined by the distribution of the random weights. For each of these three phases, we give the expression of the free energy per unit length in the limit of infinitely long polymers. Our proofs require several hypotheses on the distribution , most importantly, that the amplitude and the phase of each complex weight be statistically independent. The main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy. We illustrate our results by some examples and discuss possible generalizations to a larger class of distributions, to Random Energy Models, and to the finite dimensional case. We note that our results are not in agreement with the predictions of a recent replica approach to a similar problem.     

Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on ℤ ₊ ^d where to each edgee is associated a random variable v(e). A polymer configuration is represented by a directed pathr and has a weight exp[-β ∑ _e ∈r ν(e)], withβ=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations     

A note on the diffusion of directed polymers in a random environment

A simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford3 and if the disorder is small enough. This simplifies and extends an approach by J. Imbrie and T. Spencer.     

Improved bounds for the transition temperature of directed polymers in a finite-dimensional random medium

We consider the problem of directed polymers in a random medium of a finitedimensional lattice. In the high-temperature phase of this system it is known that the annealed and quenched free energies coincide. Upper bounds on the transition temperature to a low-temperature phase had previously been obtained by calculating the first two moments Z and Z² of the partition function. We improve these bounds by estimating noninteger moments Z for 1<<2.     

Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media

The correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media was analytically derived. We showed the correlation depends on spatial Fourier transforms of both the intensity and degree of spatial correlation of scattering potentials of the media, while the normalized correlation equals the squared modulus of the degree of spatial coherence of the scattered fields.     

Lyapunov exponents of large,sparse random matrices and the problem of directed polymers with complex random weights

We present results on two different problems: the Lyapunov exponent of large, sparse random matrices and the problem of polymers on a Cayley tree with random complex weights. We give an analytic expression for the largest Lyapunov exponent of products of random sparse matrices, with random elements located at random positions in the matrix. This expression is obtained through an analogy with the problem of random directed polymers on a Cayley tree (i.e., in the mean field limit), which itself can be solved using its relationship with random energy models (REM and GREM). For the random polymer problem with complex weights we find that, in addition to the high- and the low-temperature phases which were already known in the case of positive weights, the mean field theory predicts a new phase (phase III) which is dominated by interference effects.     

Multiple scattering in random media. III. Coherent potential propagators and fluctuations

An earlier microscopic approach to the theory of the averaged resolvent operator for an electron interacting with impurities is formulated in terms of coherent propagators. We study the corrections to the coherent potential approximation arising from fluctuations. For uncorrelated positions of the impurities, the linear, restricted, and general two-body additive approximations to the treatments of fluctuations are studied. For general correlations, the linear and restricted two-body additive approximations are studied. For both coherent and bare propagators, corresponding treatments of fluctuations involve the same correlation functions for impurities.Work supported in part by the National Science Foundation under Contract No. NSF DMR 79-23213.