and their merge is associated with a formation of junctions over saddle points [@yastrebov2014tl; @dapp2015epl], which cannot be described in the framework of Hertzian theory which assumes the parabolic shape of contacting solids. Rigorously in the framework of statistical model we shall assume that at least the contact radius associated with asperities does not overpass the quarter[^2] of the average distance between asperities $\langle d\rangle$ (see \[app\] for details), i.e. $${a_{\mbox{\tiny micro}}}< \frac{\langle d\rangle}{4} = \frac{1}{4\sqrt D} \approx \lambda_l \sqrt{\frac{\sqrt3 }{8  \pi } \frac{(2-H)}{(1-H)}\frac{(\zeta^{2-2H} - 1)}{\zeta^{4-2H}}}   \label{amicro_limit}$$ where $D$ is the density of asperities, and the last approximate equality is valid for surfaces with a rich spectral content, i.e. for $\zeta = \lambda_l/\lambda_s \gg 1$.

In most statistical models, both multi-asperity and Persson’s model, the rough surface is assumed to be Gaussian, i.e. the heights have an infinite support. A notable exception is the study of asperity-height distribution on a semi-infinite support [@ciavarella2017asme]. Therefore, for any reasonable approaching (here, the distance between mean planes of rough surfaces expressed in roughness-dispersion units), a small but non-zero portion of the surface will experience an infinite displacement, which would readily result in an apparent violation of criterion . However, since in statistical models 