On the Field Dependence of Random Walks in the Presence of Random Fields (original) (raw)
Random Walk in a Random Environment and 1f Noise
Physical Review Letters, 1983
Here we argue that in this case the power spec-trum behaves as ln'f/f for small f. The argument runs as follows: The random force acting at x derives from a random potential V such that potential differences scale like A,''when distances are multiplied by~. The dynamics is dominated by the long time it takes to cross" mountains"(moutain passes if we were in several dimensions). When the particle is con-fined in a valley it has equilibrium distribution-exp (-V). When distances are multiplied by A., the corresponding transition ...
Random walks with short-range interaction and mean-field behavior
Journal of Statistical Physics, 1994
We introduce a model of self-repelling random walks where the short-range interaction between two elements of the chain decreases as a power of the difference in proper time. Analytic results on the exponent ν are obtained. They are in good agreement with Monte Carlo simulations in two dimensions. A numerical study of the scaling functions and of the efficiency of the algorithm is also presented.
Annales de l Institut Henri Poincaré Probabilités et Statistiques
We consider, as in part I (see above), a random walk X t ∈ℤ ν , t∈ℤ + , and a dynamical random field ξ t (x), x∈ℤ ν , in mutual interaction with each other. The model is a perturbation of an unperturbed model in which walk and field evolve independently. Here we consider the environment process in a frame of reference that moves with the walk, i.e., the “field from the point of view of the particle” η t (·)=ξ t (X t +·). We prove that its distribution tends, as t→∞, to a limiting distribution μ, which is absolutely continuous with respect to the unperturbed equilibrium distribution. We also prove that, for ν≥3, the time correlations of the field η t decay as const·e -αt /t ν/2 .
Random walk in a one-dimensional random medium
Physica A: Statistical Mechanics and its Applications, 1990
1"he random walk of a particle in a one-dimensional random mcdmm is examined by means of the cquiwflent transfer rates technique, in the discrcle as well as m the continuous version of lhe model.
Biased random walk on a biased random walk
Physica A: Statistical Mechanics and its Applications, 1991
We consider the random walk of a particle along topologically linear channels under the influence of a uniform drift force. The channels are generated by the usual biased random walk procedure. The resulting mean-and mean-square displacements of a particle are discussed.
Interacting random walk in a dynamical random environment. I: Decay of correlations
Annales De L Institut Henri Poincare-probabilites Et Statistiques, 1994
We consider a random walk Xt, t E Z+ and a dynamical random Geld ~ (x ) , x E (t E Z+) in mutual interaction with each other. The interaction is small, and the model is a perturbation of an unperturbed model in which walk and field evolve independently, the walk according to i.i.d. finite range jumps, and the field independently at each site x E llv, according to an ergodic Markov chain. Our main result in Part I concerns the asymptotics of temporal correlations of the random field, as seen in a fixed frame of reference. We prove that it has a "long time tail" falling off as an inverse power of t. In Part II we obtain results on temporal correlation in a frame of reference moving with the walk. (*) Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds. A.M.S. Classification : 60 J 15, 60 J 10. Annales de l’lnstitut Henri Poincaré Probabilités et Statistiques 0246-0203 Vol. 30/94/04/$ 4.00/@ Gauthier-Villars 520 C. BOLDRIGHINI, R. A. MINLOS AND A. PELLEGRIN...
Biased random walks and propagation failure
2007
The critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non-Markovian biased random walks with reactions. From the Hamilton-Jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation.
Current fluctuations of a system of one-dimensional random walks in random environment
Annals of Probability, 2010
We study the current of particles that move independently in a common static random environment on the one-dimensional integer lattice. A twolevel fluctuation picture appears. On the central limit scale the quenched mean of the current process converges to a Brownian motion. On a smaller scale the current process centered at its quenched mean converges to a mixture of Gaussian processes. These Gaussian processes are similar to those arising from classical random walks, but the environment makes itself felt through an additional Brownian random shift in the spatial argument of the limiting current process.
Two Results on Asymptotic Behaviour of Random Walks in Random Environment
2016
Two results on Asymptotic Behaviour of Random Walks in Random Environment Jeremy Voltz Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2016 In the first chapter of this thesis, we consider a model of directed polymer in 1 + 1 dimensions in a product-type random environment ω(t, x) = b(t)F (x), where the fields F and b are i.i.d., with F (x) continuous, symmetric and bounded, and b(t) = ±1 with probabilty 1/2. Thus ω can be viewed as the field F oscillating in time. We consider directed last-passage percolation through this random field; namely, we investigate the behavior of the length n polymer path with maximal action, where the action of a path is simply the sum of the environment variables it moves through. We prove a law of large numbers for the maximal action of the path from the origin to a fixed endpoint (n, bαnc), and investigate the limiting shape function a(α). We prove that this shape function is non-linear, and has a corner at α = 0, thus i...