This talk is based on joint work with Pavel Bleher and Ken McLaughlin (arXiv:2112.09412) where we study the topological expansion in the complex quartic random matrix model in the absence of the cubic term. I will talk about the following new results which have been obtained using the theory of quadratic differentials and the Riemann-Hilbert analysis for the complex quartic matrix model:

1. Analytical description of the critical contours in the complex parameter plane where phase transitions between the one-cut, two-cut, and the three-cut regimes in this quartic model take place, thereby proving a result of Francois David [Phases of the large-n matrix model and non-perturbative effects in 2d gravity. Nuclear Physics B, 348(3):507-524, 1991].

2. The $1/N^2$ expansion for the free energy in the entire one-cut regime of the complex parameter.

3. The precise information about the generating functions of the topological expansion (and extension of some of the results in the celebrated paper of Bessis, Itzykson, and Zuber [Quantum field theory techniques in graphical enumeration. Advances in Applied Mathematics, 1(2):109 - 157, 1980.]. This allows us to explicitly find the number $N_j(g)$ of 4-valent connected labeled graphs with j vertices on a compact Riemann surface of genus g. I will also talk about the leading order asymptotics of $N_j(g)$ for an arbitrary genus g, as the number of vertices of the 4-valent graphs tends to infinity.

If time permits, I will also discuss the openness of the q-cut regular regime in the presence of a general complex polynomial external field. Speaker(s): Roozbeh Gharakhloo (Colorado State University)