From discrete surfaces and matrix models to topological recursions IV
by Nicholas Orantin
Abstract: This lecture shows how a Riemann surface arises from a matrix model as a branched cover with known periods of a holomorphic differential
over the Riemann surface. Random matrix models represent a wonderful
tool in the enumeration of random discrete surfaces of given topology.
These lectures will address three issues. I will first properly define the concept of formal matrix integral used to build generating functions of discrete surfaces. I will then show that the enumeration of all possible ways of removing one edge from such a surface gives a set of loop equations which can be solved by induction in terms of an algebraic curve characterizing the considered matrix model: the spectral curve.
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