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A Quick Derivation of the Loop Equations for Random Matrices

N. M. Ercolani·K. D. T-R McLaughlin
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Abstract

The "loop equations" of random matrix theory are a hierarchy of equations born of attempts to obtain explicit formulae for generating functions of map enumeration problems. These equations, originating in the physics of 2-dimensional quantum gravity, have lacked mathematical justification. The goal of this paper is to provide a complete and short proof, relying on a recently established complete asymptotic expansion for the random matrix theory partition function.

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N. M. Ercolani, K. D. T-R McLaughlin. 2006-09-18. A Quick Derivation of the Loop Equations for Random Matrices. https://arxiv.org/abs/math-ph/0609048

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