Average persistence in random walks

Abstract

We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P_pr(t) that the expectation value <x(t)> of the walker's position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a logarithmic dependence P_pr(t)~{ln(t)}^theta' with a new exponent theta'=0.191+/-0.002. We note on a complete correspondence between the average persistence of random walks and the magnetization autocorrelation function of the transverse-field Ising chain, in the homogeneous and disordered case.

6 pages LaTeX, 3 postscript figures included

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• Persistence exponents for fluctuating interfaces
• Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-dimensional Random Environments
• The Random Transverse Ising Spin Chain and Random Walks
• Non-Markovian Persistence and Nonequilibrium Critical Dynamics
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• Persistence and First-Passage Properties in Non-equilibrium Systems
• Strong disorder RG approach of random systems
• Absorbing state phase transitions with quenched disorder
• Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions
• Exact Asymptotic Results for Persistence in the Sinai Model with Arbitrary Drift
• Random quantum magnets with long-range correlated disorder: Enhancement of critical and Griffiths-McCoy singularities
• Persistence of Randomly Coupled Fluctuating Interfaces
• Non equilibrium dynamics below the super-roughening transition
• Percolation in random environment
• Non Markovian persistence in the diluted Ising model at criticality
• Low-temperature dynamics of Long-Ranged Spin-Glasses : full hierarchy of relaxation times via real-space renormalization
• Probability distribution of the sizes of largest erased-loops in loop-erased random walks