Myopic self-avoiding fractional Brownian motion with a reflecting wall

Myopic self-avoiding fractional Brownian motion (FBM), also called FBM with

mean-density interactions, is the problem of a particle moving under the influence of

long-range correlated noise while trying to avoid already visited locations [1].

Here,we investigate FBM with mean-density interaction confined by a reflecting

boundary at the origin. Monte Carlo simulations reveal that the interplay of the reflect-

ing wall and the mean-density interaction qualitatively changes the process. Specif-

ically, the interaction cuts off the singularity observed in the probability density of

reflected FBM in the super diffusive regime [2]. For anomalous diffusion exponent

α >4/3, we find a fundamental breakdown of single-parameter scaling, creating two

distinct dynamical regimes separated by a time-dependent crossover length scale x_cr(t).

Near the origin (x < x_cr), the integrated probability density exhibits plateau behavior

with P_int(0, t) ∼ t^α/4. Beyond the crossover length (x > x_cr), the system recovers the

usual FBM scaling. For α < 4/3, in contrast, the process is interaction-dominated

both close to the wall and for large x. The resulting probability density follows one-

parameter scaling. We discuss consequences for applications of FBM, for example the

growth serotonergic axons in vertebrate brains [3].