KPZ-like scaling on a high-dimensional hypersphere

We consider the orientational diffusion controlled by the hyperspherical Laplacian, ${\mathrm{\nabla }}_D^2$, on the surface of the $D$–dimensional hypersphere in the limit $D\to \mathrm{\infty }$. We find that for stretched paths with lengths relatively short compared to the hypersphere’s radius, the finite-size corrections in orientational correlations are controlled by the Kardar-Parisi-Zhang (KPZ) scaling exponent, $\gamma =1\mathrm{/}3$. In addition, we speculate about the topology of the orientational target space representing the surface of the hypersphere.