Verification of KPZ-class variance exponents

Prove the predicted superdiffusive variance growth orders \(t^{4/3}\) in one dimension and \(t(\log t)^{2/3}\) in two dimensions for the occupation-time functional of asymmetric exclusion at density \(\rho\neq 1/2\).

Background

For asymmetric exclusion with nonzero drift, the notes establish finite-variance behavior away from density one-half in sufficiently high dimensions and lower bounds at the critical density. They state that the expected orders in dimensions one and two are associated with KPZ-class phenomena.

These predicted exponents are motivated by the conjectural scaling of the return probability of a second-class particle and a Gaussian ansatz relating that probability to the variance of the particle’s displacement. A rigorous verification of the resulting occupation-time variance orders remains unresolved.