Caustics, counting maps and semi-classical asymptotics - Mathematical PhysicsReport as inadecuate

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Abstract: This paper develops a deeper understanding of the structure and combinatorialsignificance of the partition function for Hermitian random matrices. Thecoefficients of the large N expansion of the logarithm of this partitionfunction,also known as the genus expansion, and its derivatives aregenerating functions for a variety of graphical enumeration problems. The mainresults are to prove that these generating functions are in fact specificrational functions of a distinguished irrational algebraic function of thegenerating function parameters. This distinguished function is itself thegenerating function for the Catalan numbers or generalized Catalan numbers,depending on the choice of parameter. It is also a solution of the inviscidBurgers equation for certain initial data. The shock formation, or caustic, ofthe Burgers characteristic solution is directly related to the poles of therational forms of the generating functions.These results in turn provide new information about the asymptotics ofrecurrence coefficients for orthogonal polynomials with respect to exponentialweights. One gains new insights into the relation between certain derivativesof the genus expansion and the asymptotic expansion of the first Painlevetranscendent, related to the double-scaling limit. This work provides a preciseexpression of the Painleve asymptotic coefficients directly in terms of thecoefficients of the partial fractions expansion of the rational form of thegenerating functions established here. Moreover, these insights point toward amore general program relating the first Painleve hierarchy and the higher orderstructure of the double-scaling limit to the specific rational structure ofgenerating functions.

Author: N. M. Ercolani



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