Matrix realization of random surfaces

The large-$N$ one-matrix model with a potential $V\left(\phi \right)=\frac{{\phi }^2}{2}+\frac{g_4{\phi }^4}{N}+\frac{g_6{\phi }^6}{N^2}$ is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients $r_k$ ($k=1, 2, 3, \dots$) of the orthogonal polynomials at large $N$. We find that for $\frac{g_6}{g_4^2}>\frac{1}{2}$ there is no $m=2\mathrm{}$ solution which can be expressed as a smooth function of $\frac{k}{N}$ in the limit $N\to \infty$. This means that the assumption of smoothness of $r_k$ at $N\to \infty$ 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 $m=2\mathrm{}$ are complementary to previous analyses by the saddle-point method. On the other hand, for the case $m=3\mathrm{}$ ($\frac{g_6}{g_4^2}=\frac{4}{5}$), we find a well-behaved solution which coincides with the result obtained by Brézin, Marinari, and Parisi. To strengthen the validity of our numerical scheme, we present in an appendix a nonperturbative solution for $m=1\mathrm{}$ which obeys the so-called type-II string equation.