Matrix Integrals and Random Planar Maps

Matrix integrals provide a powerful framework for solving complex problems at the intersection of mathematical physics and combinatorics. This course offers a clear, structured path through the mathematical foundations connecting matrix models, planar graph enumeration, and statistical mechanics models. You will explore foundational concepts from the ground up, starting with core terminology, Gaussian matrix integrals, and combinatorial graph expansion, before examining the solvable Potts model on random planar surfaces. What you will learn: Understand the core principles and algebraic foundations of random matrix models. Formulate and solve matrix integrals using Wick's theorem and diagrammatic expansions. Explore the combinatorial properties of planar maps embedded on surfaces. Analyze the Potts model and random graph colorings using matrix formalisms. Apply modern analytical techniques to investigate phase transitions and critical phenomena. The course begins with elementary definitions and mathematical prerequisites, progressively advancing through detailed theoretical derivations and worked analytical examples. This guide is designed for learners with an interest in statistical mechanics, theoretical physics, or discrete mathematics who want an accessible entry point to matrix models without heavy prerequisites. Start reading today to deepen your understanding of modern mathematical physics.