Integration over positive matrices: Comparative analysis of the Penner model and its modification

We construct a positive matrix model by modifying the Penner potential. Solving such a model exactly both at the lattice level and in the continuum limit, we find that it shares with the Penner model the same free energy in the double scaling limit and thus resembles the theory of strings at c=1. A further comparison shows that the continuum one-point function of our model is different from that of the Penner model as well as c=1 strings, at least on the torus. This opens up the interesting possibility of approaching some non-(p, q) strings via integration over positive random matrices. The result may also raise complications associated with the physical identification of the Penner model. ((orig.))