  Our main result is that for densities $<\frac{3}{10}$ a random group in the
square model has the Haagerup property and is residually finite. Moreover, we
generalize the Isoperimetric Inequality, to some class of non-planar diagrams
and, using this, we introduce a system of modified hypergraphs providing the
structure of a space with walls on the Cayley complex of a random group. Then
we show that the natural action of a random group on this space with walls is
proper, which gives the proper action of a random group on a CAT(0) cube
complex.
