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A new approach to solving random matrix models directly in the

large N limit is developed. First, a set of numerical values for some

low-pt correlation functions is guessed. The large N loop equations are

then used to generate values of higher-pt correlation functions based on

this guess. Then one tests whether these higher-pt functions are

consistent with positivity requirements, e.g., tr M^{2k} > 0. If not,

the guessed values are systematically ruled out. In this way, one can

constrain the correlation functions of random matrices to a tiny

subregion which contains (and perhaps converges to) the true solution.

This approach is tested on single and multi-matrix models and handily

reproduces known solutions. It also produces strong results for

multi-matrix models which are not believed to be solvable. A tantalizing

possibility is that this method could be used to search for new critical

points, or string worldsheet theories.