  The hypersphere model is a simple one-parameter model of the potential energy
landscape of viscous liquids, which consists of a percolating system of
hyperspheres of equal sizes randomly distributed in $R^{3N}$ where $N$ is the
number of particles. We study random walks within overlapping hyperspheres in
12 to 45 dimensions, utilizing an algorithm for on-the-fly placement of the
hyperspheres in conjunction with the kinetic Monte Carlo method. We find
behavior typical of viscous liquids; thus decreasing the hypersphere density
(corresponding to decreasing the temperature) leads to a slowing down of the
dynamics by many orders of magnitude. The shape of the mean-square displacement
as a function of time is found to be very similar to that of the Kob-Andersen
binary Lennard-Jones mixture and the Random Barrier Model, which predicts well
the frequency-dependent fluidity of nine glass-forming liquids of different
chemistry [Bierwirth et al., Phys. Rev. Lett. 119, 248001 (2017)].
