Restoring the fluctuation-dissipation theorem in Kardar-Parisi-Zhang universality class through a new emergent fractal dimension

The Kardar-Parisi-Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation-dissipation theorem (FDT) for spatial dimension $d>1$. Notably, these questions were answered exactly only for $1+1$ dimensions. In this work, we propose a new FDT valid for the KPZ problem in $d+1$ dimensions. This is done by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension $d_n$. We present relations between the KPZ exponents and two emergent fractal dimensions, namely $d_f$, of the rough interface, and $d_n$. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent $α$, the surface fractal dimension $d_f$ and, through our relations, the noise fractal dimension $d_n$. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, a FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.