Multi – matrix models without continuum limit

We derive the discrete linear systems associated to multi–matrix models, the corresponding discrete hierarchies and the appropriate coupling conditions. We also obtain the W 1+∞ constraints on the partition function. We then apply to multi– matrix models the technique, developed in previous papers, of extracting hierarchies of differential equations from lattice ones without passing through a continuum limit. In a q–matrix model we find 2q coupled differential systems. The corresponding differential hierarchies are particular versions of the KP hierarchy. We show that the multi–matrix partition function is a τ –function of these hierarchies. We discuss a few examples in the dispersionless limit.