  We consider the matrix model approach to the anomalous dimension matrix in
$\mathcal{N}=4$ super Yang--Mills theory. We construct the path integral
representation for the anomalous dimension density matrix and analyze the
resulting action. In particular, we consider the large $N$ limit, which results
in a classical field theory. Since the same limit leads to spin chains, we
propose to consider the former as an alternative description of the latter. We
consider also the limit of small $N$, which corresponds to the restriction to
the diagrams of maximal topological genus.
