Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system come at two levels. On the central limit scale n1/2 one sees initial fluctuations transported along characteristics and no dynamical noise. The second order of fluctuations comes from the particle current across the characteristic. For a system made up of independent random walks we show that the second-order fluctuations appear at scale n1/4 and converge to a certain self-similar Gaussian process. If the system is in equilibrium, this limiting process specializes to fractional Brownian motion with Hurst parameter 1/4. This contrasts with asymmetric exclusion and Hammersley’s process whose second-order fluctuations appear at scale n1/3, as has been discovered through related combinatorial growth models.
"Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks." Ann. Probab. 33 (2) 759 - 797, March 2005. https://doi.org/10.1214/009117904000000946