Random Walkers in 1-D Random Environments: Exact Renormalization Group Analysis

Abstract

Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position $x(t)$ of a particle, the probability of it not returning to the origin and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite size sample. We compute the distribution of meeting time of two particles. We also obtain a detailed analytic description of thermally averaged trajectories: we compute the distributions of the number of returns and of the number of jumps forward. They obey multifractal scaling, characterized by generalized persistence exponents $\theta (g)$ which we compute. With a small bias, the number of returns is finite, characterized by a universal scaling function. The statistics of the successive times of return of thermally averaged trajectories is obtained. The two time distribution of the positions of a particle, $x(t)$ and $x(t^{\prime })$ ($t>t^{\prime }$) is computed exactly. It exhibits ``aging'' with several regimes: without a bias, for $t-t^{\prime }\sim t^{\prime \alpha },\alpha >1$, it exhibits a $(\ln t)/(\ln t^{\prime })$ scaling, with a novel singularity at rescaled positions $x(t)=x(t^{\prime })$. For closer times $\alpha <1$ there is a quasi-equilibrium regime with $\ln (t-t^{\prime })/\ln t^{\prime }$ scaling. The crossover to a $t/t^{\prime }$ aging form under a small bias is obtained analytically. Rare events, e.g. splitting of the thermal packet between wells, are also studied. Connections with the Green's function of a 1D Schrödinger problem and quantum spin chains are discussed.

81 pages, Latex. One subsection modified, one added

References in corpus (3)

• Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-dimensional Random Environments
• Percolation Transition in the random antiferromagnetic spin-1 chain
• Aging and diffusion in low dimensional environments

Cited by in corpus (130)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Rare region effects at classical, quantum, and non-equilibrium phase transitions
• Strong disorder RG approach of random systems
• Entanglement entropy of random quantum critical points in one dimension
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Pulling Pinned Polymers and Unzipping DNA
• Random walks with preferential relocations to places visited in the past and their application to biology
• DNA unzipped under a constant force exhibits multiple metastable intermediates
• Single Molecule Statistics and the Polynucleotide Unzipping Transition
• Aging Continuous Time Random Walks
• Fractional Fokker-Planck Equation for Ultraslow Kinetics
• Separation of time and length scales in spin-glasses: temperature as a microscope
• Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes
• Absorbing state phase transitions with quenched disorder
• Efficiency, selectivity and robustness of the nuclear pore complex transport
• Renormalization group study of the two-dimensional random transverse-field Ising model
• Entanglement entropy dynamics of disordered quantum spin chains
• Exact solutions for the statistics of extrema of some random 1D landscapes, Application to the equilibrium and the dynamics of the toy model
• Nonequilibrium dynamics of random field Ising spin chains: exact results via real space RG
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Integrable 2D Lorentzian Gravity and Random Walks
• Hopping in the Glass Configuration Space: Subaging and Generalized Scaling Laws
• Partially asymmetric exclusion models with quenched disorder
• Loop Dynamics in DNA Denaturation
• Functionals of the Brownian motion, localization and metric graphs
• Localization and universal fluctuations in ultraslow diffusion processes
• Entanglement entropy of the random spin-1 Heisenberg chain
• Anomalous diffusion, Localization, Aging and Sub-aging effects in trap models at very low temperature
• Fluctuations of time averages for Langevin dynamics in a binding force field
• Scaling Green-Kubo relation and application to three aging systems
• Velocity and diffusion constant of an active particle in a one dimensional force field
• Partially Asymmetric Exclusion Processes with Sitewise Disorder
• Localization of thermal packets and metastable states in Sinai model
• Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model
• Localization properties of the anomalous diffusion phase $x{t}^{\mu }$ in the directed trap model and in the Sinai diffusion with bias
• Non-linear Response of the trap model in the aging regime : Exact results in the strong disorder limit
• Linear and non linear response in the aging regime of the 1D trap model
• Reaction Diffusion Models in One Dimension with Disorder
• Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets
• Strong Griffiths singularities in random systems and their relation to extreme value statistics
• Aperiodic quantum XXZ chains: Renormalization-group results
• Optimization and universality of Brownian search in quenched heterogeneous media
• Exact Asymptotic Results for Persistence in the Sinai Model with Arbitrary Drift
• Langevin simulations of the out-of-equilibrium dynamics of the vortex glass in high-temperature superconductors
• The partially asymmetric zero range process with quenched disorder
• Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials
• Freezing of dynamical exponents in low dimensional random media
• Persistent Sinai type diffusion in Gaussian random potentials with decaying spatial correlations
• Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems
• Mean first-passage and residence times of random walks on asymmetric disordered chains
• Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder
• Finite-size scaling properties of random transverse-field Ising chains : Comparison between canonical and microcanonical ensembles for the disorder
• Ramifications of disorder on active particles in one dimension
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Energy dynamics in the Sinai model
• A numerical approach to copolymers at selective interfaces
• Persistence of Randomly Coupled Fluctuating Interfaces
• Moderate deviations for diffusions with Brownian potentials
• Finite-range viscoelastic subdiffusion in disordered systems with inclusion of inertial effects
• Random walks and polymers in the presence of quenched disorder
• Sinai model in presence of dilute absorbers
• Fluctuations of random matrix products and 1D Dirac equation with random mass
• One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
• One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
• Duality symmetries and effective dynamics in disordered hopping models
• Persistence of a Continuous Stochastic Process with Discrete-Time Sampling: Non-Markov Processes
• Supersymmetric quantum mechanics with Levy disorder in one dimension
• Survival and residence times in disordered chains with bias
• Anomalous Fluctuations of Extremes in Many-Particle Diffusion
• Non equilibrium dynamics of disordered systems : understanding the broad continuum of relevant time scales via a strong-disorder RG in configuration space
• Effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian
• Spectral analysis of Sinai's walk for small eigenvalues
• Random elastic networks : strong disorder renormalization approach
• Aging properties of Sinai's model of random walk in random environment
• On a non-linear Fluctuation Theorem for the aging dynamics of disordered trap models
• Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses
• Persistence in Brownian motion of an ellipsoidal particle in two dimensions
• Dynamic fluctuations of elastic lines in random environments
• Equilibrium of disordered systems : constructing the appropriate valleys in each sample via strong disorder renormalization in configuration space
• Low-temperature properties of some disordered systems from the statistical properties of nearly degenerate two-level excitations
• Exact Meander Asymptotics: a Numerical Check
• Breaking supersymmetry in a one-dimensional random Hamiltonian
• Unzipping Vortices in Type-II Superconductors
• Strong Disorder Renewal Approach to DNA denaturation and wetting : typical and large deviation properties of the free energy
• Non-equilibrium dynamics of polymers and interfaces in random media : conjecture $\psi =d_s/2$ for the barrier exponent
• Strong Disorder Renormalization for the dynamics of Many-Body-Localized systems : iterative elimination of the fastest degree of freedom via the Floquet expansion
• Stochastic Aggregation: Scaling Properties
• Competition between quenched disorder and long-range connections: A numerical study of diffusion
• Persistence in Random Walk in Composite Media
• DNA unzipping and the unbinding of directed polymers in a random media
• Random walk in two-dimensional self-affine random potentials : strong disorder renormalization approach
• Time correlations and persistence probability of a Brownian particle in a shear flow
• Non-equilibrium dynamics of finite-dimensional disordered systems : RG flow towards an "infinite disorder" fixed point at large times
• Temperature shifts in the Sinai model: static and dynamical effects
• Slow relaxation due to optimization and restructuring: Solution on a hierarchical lattice
• Driven interfaces in random media at finite temperature : is there an anomalous zero-velocity phase at small external force ?
• Random field disorder at an absorbing state transition in one and two dimensions
• Duality between random trap and barrier models
• Local equilibrium properties of ultraslow diffusion in the Sinai model
• Aging Feynman-Kac Equation
• Random walks and branching processes in correlated Gaussian environment
• Almost sure asymptotics for a diffusion process in a drifted Brownian potential
• Entanglement between random and clean quantum spin chains
• Random walk models and probabilistic techniques for inhomogeneous polymer chains
• Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments
• First passage time distribution for a random walker on a random forcing energy landscape
• Random walks in a random environment on a strip: a renormalization group approach
• Anomalous transport in disordered exclusion processes with coupled particles
• Dynamical barriers for the random ferromagnetic Ising model on the Cayley tree : traveling-wave solution of the real space renormalization flow
• Anomalous diffusion in disordered multi-channel systems
• Scaling of local persistence in the disordered contact process
• Entanglement across extended random defects in the XX spin chain
• Diffusion in random environment and the renewal theorem
• The generalized Lyapunov exponent for the one-dimensional Schrödinger equation with Cauchy disorder: some exact results
• Random Walk on a Random Surface: Implications of Non-perturbative Concepts and Dynamical Emergence of Galilean Symmetry
• Subdiffusive motion in kinetically constrained models
• Infinite disorder renormalization fixed point for the continuum random field Ising chain
• Diffusion between evolving interfaces
• Random walk in a two-dimensional self-affine random potential : properties of the anomalous diffusion phase at small external force
• Random wetting transition on the Cayley tree : a disordered first-order transition with two correlation length exponents
• Renormalization-group investigation of the S=1 random antiferromagnetic Heisenberg Chain
• First-passage on disordered intervals
• Law of large numbers for a finite-range random walk in a dynamic random environment with nonuniform mixing
• Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension
• The alternating marked point process of h-slopes of the drifted Brownian motion
• Localization of favorite points for diffusion in random environment
• Finite current stationary states of random walks on one-dimensional lattices with aperiodic disorder
• On constrained annealed bounds for pinning and wetting models
• Metastable states in Brownian energy landscape
• Methods for detecting Order-by-Disorder transitions: the example of the Domino model