Integrable Systems and Partition Functions of Random Matrix Models - Pinheiro, Carla Mariana da Silva (2022)
Random Matrix Theory is a hot topic nowadays given its wide range of applications in different areas, such as quantum mechanics, machine learning, dynamical systems, among others. The present work begins reviewing some of the most famous applications. Then, particular attention is given to the enumeration of maps through the expectation of the trace of random matrices in a Gaussian Unitary Ensemble. Latter, an asymptotic expansion of the partition function is developed, which allows one to count maps by the connection between the expectation of the trace and the partition function. Such expansion is explored in full details and the calculations involving an important RiemannHilbert problem are explicitly worked out. At last, connections between random matrices and integrable systems are explored in two different ways. When the dimension is fixed, the partition function of a random matrix model is a tau-function of the KP hierarchy, while in the limit where the dimension goes to infinity one recovers Painlevé solutions.