MATRIX REALIZATION OF RANDOM SURFACES

The large-N one-matrix model with a potential V (phi) = phi-2/2 + g4-phi-4/N + g6-phi-6/N2 is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients r(k) (k = 1,2,3,...) of the orthogonal polynomials at large N. We find that for g6/g4(2) > 1/2 there is no m = 2 solution which can be expressed as a smooth function of k/N in the limit N --> infinity. This means that the assumption of smoothness of r(k) at N --> infinity near the critical point, which was essential to derive the string susceptibility and the string equation, is broken even at the tree level of the genus expansion by adding the phi-6 term. We have also observed the free energy around the (expected) critical point to confirm that the system does not have the desired criticality as pure gravity. Our (discouraging) results for m = 2 are complementary to previous analyses by the saddle-point method. On the other hand, for the case m = 3 (g6/g4(2) = 4/5), we find a well-behaved solution which coincides with the result obtained by Brezin, Marinari, and Parisi. To strengthen the validity of our numerical scheme, we present in an appendix a nonperturbative solution for m = 1 which obeys the so-called type-II string equation.