Date: Monday, March 07, 2022
Location: ZOOM ID: 926 6491 9790 Virtual (4:00 PM to 5:00 PM)
Title: Phase Diagram and Topological Expansion in the Complex Quartic Random Matrix Model
Abstract: 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 RiemannHilbert analysis for the complex quartic matrix model:
1. Analytical description of the critical contours in the complex parameter plane where phase transitions between the onecut, twocut, and the threecut regimes in this quartic model take place, thereby proving a result of Francois David [Phases of the largen matrix model and nonperturbative effects in 2d gravity. Nuclear Physics B, 348(3):507524, 1991].
2. The $1/N^2$ expansion for the free energy in the entire onecut 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 4valent 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 4valent graphs tends to infinity.
If time permits, I will also discuss the openness of the qcut regular regime in the presence of a general complex polynomial external field.
A recording of the talk can be found here.
Files:
Speaker: Roozbeh Gharakhloo
Institution: Colorado State University
Event Organizer: Ahmad Barhoumi barhoumi@umich.edu
