Phase Diagram and Topological Expansion in the Complex Quartic Random Matrix Model

We use the Riemann-Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers $\mathscr{N}_j(g)$ of $4$-valent connected graphs with $j$ vertices on a compact Riemann surface of genus $g$. We explicitly evaluate these numbers for Riemann surfaces of genus $0,1,2,$ and $3$. Also, for a Riemann surface of an arbitrary genus $g$, we also calculate the leading term in the asymptotics of $\mathscr{N}_j(g)$ as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David \cite{DAVID}. These phase transitions are of the following four types: a) one-cut to two-cut through the splitting of the cut at the origin, b) two-cut to three-cut through the birth of a new cut at the origin, c) one-cut to three-cut through the splitting of the cut at two symmetric points, and d) one-cut to three-cut through the birth of two symmetric cuts.