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In the context of the Integer Quantum Hall plateau transitions, we formulatea specific map from random landscape potentials onto 2D discrete randomsurfaces. Critical points of the potential, namely maxima, minima and saddlepoints uniquely define a discrete surface $S$ and its dual $S^*$ made ofquadrangular and $n-$gonal faces, respectively, thereby linking the geometry ofthe potential with the geometry of discrete surfaces. The map isparameter-dependent on the Fermi level. Edge states of Fermi lakes moving alongequipotential contours between neighbour saddle points form a network ofscatterings, which define the geometric basis, in the fermionic model, for theplateau transitions. The replacement probability characterizing the networkmodel with geometric disorder recently proposed by Gruzberg, Kl\"umper, Nudingand Sedrakyan, is physically interpreted within the current framework as aparameter connected with the Fermi level.

Geometry of random potentials: Induction of 2D gravity in Quantum Hall plateau transitions