Abstract
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 M2k) ≥ 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.
Highlights
In this paper, we propose a method to solve multi-matrix models in the strict large N limit
The large N loop equations are used to generate values of higher-pt correlation functions based on this guess
We propose a method to solve multi-matrix models in the strict large N limit
Summary
The loop equations relate higher-pt functions to lower-pt functions. Since the number of correlation functions of fixed degree is growing exponentially for even a 2-matrix model, it may not seem obvious that s∗ should even be finite. We will argue that if we know all correlation functions of degree at most k∗, the loop equations determine the rest. The number of loop equations for correlators of degree k is ∼ mk. We expect that when k gets large enough so that mk ∼ mk+D/(k + D), we will have enough equations to determine the rest of the correlators For the simple case m = 2, D = 3, direct calculation gives s∗ ≤ 5, e.g. knowing the 5 traces A, B, A2, AB, B2 is enough to determine the rest of the correlation functions. An obvious question for future work is to understand if there is a simple criteria for calculating s∗ for a polynomial interaction in the matrices.
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