  We systematically check explicit and implicit assumptions of Persson's
contact mechanics theory. It casts the evolution of the pressure distribution
${\rm Pr}(p)$ with increasing resolution of surface roughness as a diffusive
process, in which resolution plays the role of time. The tested key assumptions
of the theory are: (a) the diffusion coefficient is independent of pressure
$p$, (b) the diffusion process is drift-free at any value of $p$, (c) the point
$p=0$ acts as an absorbing barrier, i.e., once a point falls out of contact, it
never reenters again, (d) the Fourier component of the elastic energy is only
populated if the appropriate wave vector is resolved, and (e) it no longer
changes when even smaller wavelengths are resolved. Using high-resolution
numerical simulations, we quantify deviations from these approximations and
find quite significant discrepancies in some cases. For example, the drift
becomes substantial for small values of $p$, which typically represent points
in real space close to a contact line. On the other hand, there is a
significant flux of points reentering contact. These and other identified
deviations cancel each other to a large degree, resulting in an overall
excellent description for contact area, contact geometry, and gap distribution
functions. Similar fortuitous error cancellations cannot be guaranteed under
different circumstances, for instance when investigating rubber friction. The
results of the simulations may provide guidelines for a systematic improvement
of the theory.
