Affiliation:
1. Department of Mathematical Sciences Indiana University ‐ Purdue University Indianapolis Indianapolis Indiana USA
2. Department of Mathematics Colorado State University Fort Collins Colorado USA
Abstract
AbstractWe 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 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 calculate the leading term in the asymptotics of 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. 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.
Funder
National Science Foundation
Simons Foundation
Subject
Applied Mathematics,General Mathematics
Reference48 articles.
1. The Map Asymptotics Constant $t_g$
2. Oper. Theory Adv. Appl;Bertola M.,2022