Motion of a random walker in a quenched power law correlated velocity field

We study the motion of a random walker in one longitudinal and d transverse dimensions with a quenched power law correlated velocity field in the longitudinal x direction. The model is a modification of the Matheron-de Marsily model, with long-range velocity correlation. For a velocity correlation function, dependent on transverse coordinates y as 1/(a+parallel to y(1)-y(2)parallel to)(alpha), we analytically calculate the two-time correlation function of the x coordinate. We find that the motion of the x coordinate is a fractional Brownian motion (FBM), with a Hurst exponent H=max[1/2,(1-alpha/4),(1-d/4)]. From this and known properties of FBM, we calculate the disorder averaged persistence probability of x(t) up to time t. We also find the lines in the parameter space of d and alpha along which there is marginal behavior. We present results of simulations which support our analytical calculation