Random Latin squares and 2-dimensional expanders

Let X be a 2-dimensional simplicial complex. The degree of an edge e is the number of 2-faces of X containing e. The complex X is an \epsilon-expander if the coboundary d_1(\phi) of every Z_2-valued 1-cochain \phi \in C^1(X;Z_2) satisfies |support(d_1(\phi))| \geq \epsilon |\supp(\phi+d_0(\psi))| for some 0-cochain \psi. Using a new model of random 2-complexes we show the existence of an infinite family of 2-dimensional \epsilon-expanders with maximum edge degree d, for some fixed \epsilon>0 and d.