the notion of asperities is hidden behind the height probability density function, it seems reasonable to find the mean local asperity radius by assuming that the contact area ${A_{\mbox{\tiny asp}}}$ predicted by a multi-asperity model over nominal area $A_0$ is formed by $N$ identical contact spots $\pi{a_{\mbox{\tiny micro}}}^2$, where the number of spots $N$ is assumed to be known through the asperity density $D$ as $N = DA_0$, therefore the individual contact-spot radius can be estimated as $${a_{\mbox{\tiny micro}}}\approx \sqrt{\frac{{A_{\mbox{\tiny asp}}}}{\pi D A_0}},$$ where from using , a simple limit for the validity of multi-asperity models could be established: $$\frac{{A_{\mbox{\tiny asp}}}}{A_0} < \frac{\pi}{16} \approx 19.6\%.$$ This limit can be used locally (point-wise) as the upper bound for multi-asperity models, i.e. the local area fraction predicted by multi-asperity model cannot overpass $19.6$ %. Of course, because of non-uniform distribution of asperities and contact spots, the real limit can be considerably smaller (see also comparison between deterministic and multi-asperity models [@yastrebov2017jmps]).

Asperities
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In vicinity of every summit, a smooth surface can be approximated by a quadratic form of local coordinates: $$z(x,y) = z_0 + a x^2 + b y^2 + 2 c xy = z_0 + {\left[\begin{array}{c}}x\\y{\end{array}\right]}^\intercal {\left[\begin{array}{cc}}a \;\;& c \\ 