scale numerical treatment of this problem, which was carried out by Pastewka & Robbins [@pastewka2016contact]. However, as remarked in [@greenwood1970contact; @pastewka2016contact], at very small loads/large separations a single-asperity contact regime exists, which results in Hertzian scaling for elastic asperities[^6].

Expectedly, our numerical study with the deterministic multi-scale model identifies these two deformation regimes (see Fig. \[fig:area\]). The data were averaged over 120 simulations per each combination of root mean squared roughness $\sigma=\{0.001,0.01,0.1\}$ and different upper cut-offs $L/\lambda_s=\{256,512\}$, the lower cut-off was kept constant $L/\lambda_l = 24$. The vertical displacement of the indenter’s tip was changed from $Z_0 = 4\sigma$ to $\sigma$ in 100 load steps, the indenter’s radius is fixed to be $R=2.5$ mm. The error bars correspond to a single standard deviation; the fact that error bars are missing in very low pressure-regime is explained by the fact that only in a single realization the contact occurs at such small loads.

The first deformation regime corresponds to the deformation of a single Hertzian asperity, it occurs at very light loads and the contact-area grows as $A_I \sim N^{2/3}$. The second regime is characterized by an approximately linear evolution of the contact area $A_{II}\sim N$ and it corresponds to multi-asperity 